Abstract
We prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain \(\Omega \) is obtained as the integral of a cost function j(u, x) depending on the solution u of a certain PDE problem on \(\Omega \). The main feature of these functionals is that the minimality of a domain \(\Omega \) cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions \(j(u,x)=-g(x)u+Q(x)\), where u is the solution of the PDE \(-\Delta u=f\) with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal u from the inwards/outwards optimality of \(\Omega \) and then we use the stability of \(\Omega \) with respect to variations with smooth vector fields in order to study the blow-up limits of the state function u. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose \(\partial \Omega \) into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of \(\partial \Omega \) we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove \(C^\infty \) regularity of the regular part of the free boundary when the data are smooth.
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1 Introduction
This paper is dedicated to the regularity of the optimal shapes, solutions to shape optimization problems of the form
where \(\mathcal {A}\) is an admissible class of open, Lebesgue measurable or quasi-open subsets of \(\mathbb {R}^d\), and where \( J:\mathcal {A}\rightarrow \mathbb {R}\) is a given shape functional, with J(A) usually depending on the solution of a PDE on the domain A. This kind of minimization problems arise in different models in Biology, Engineering and Physics (see for example [6, 21] for an overview) and have been extensively studied from both numerical and theoretical points of view. In particular, there are two classes of shape optimization problems of the form above with long history, both leading to overdetermined elliptic PDE problems with Dirichlet boundary conditions.
The first class involves the so-called spectral functionals, that is, functionals depending on the eigenvalues of the Dirichlet Laplacian as
where \(\varphi :\mathbb {R}^k\rightarrow \mathbb {R}\) is a real-valued function and |A| denotes the Lebesgue measure of A. The associated shape optimization problems
have a long history and are related to the classical question “Can one hear the shape of the drum?” and, more generally, to the interplay between the geometry of the domains and the spectrum of the Dirichlet Laplacian. The first results on the characterization of the optimal shapes, for the first and the second Dirichlet eigenvalues, go back to the works of Faber–Krahn (1922) and Krahn–Szegö (1923), and consist in finding explicit minimizers (balls and unions of disjoint balls), which is only possible in some special cases as \(J=\lambda _1\) and \(J=\lambda _2\). Today, thanks to theory developed by Buttazzo and Dal Maso [10], and to the more recent results [5, 30], it is well-known that, for monotone functionals \(\varphi \), minimizers exist in a class of measurable (quasi-open) sets. The regularity of the optimal shapes has also been extensively studied; we refer to [4] and [34] for the case of optimal sets of \(\lambda _1\) in a box, to [32] for the optimal sets of \(\lambda _2\) in a box, and to [8, 25, 26, 31] (see also [15, 32]) for functionals involving higher eigenvalues \(\lambda _k\).
The second class of functionals involves integral shape functionals, namely, for every bounded open set \(A\subset \mathbb {R}^d\) we define
where the cost function
is fixed and the state function \(u_A\) is the (weak) solution of the PDE
the force term \(f:\mathbb {R}^d\rightarrow \mathbb {R}\) being also a prescribed function.
Optimization problems for integral shape functionals arise in Optimal Control and in models from Mechanics, in which the optimization criteria \(j(u_A,x)\) takes into account external factors and forces that might appear not immediately, but only after the design is complete and the state function \(u_A\) is already fixed. This type of problems pops up also in population dynamics, when one aims to optimize the population size. As in the case of spectral functionals, also for integral functionals with monotone cost, the existence of optimal shapes in bounded domains follows from the general theory of Buttazzo and Dal Maso, and again the solutions belong to the large class of measurable (or quasi-open) sets. This existence problem was studied, in a more general framework, in [13]. A general existence result in the class of open sets, was proved recently in [11] and in [12]. Precisely, it was shown that if D is a bounded open set in \(\mathbb {R}^d\) and if the function j satisfies some suitable growth assumptions, then the shape optimization problem
has a solution \(\Omega \subset D\), \(\Omega \)-open.
On the other hand, even if the existence theory is quite well understood, there is no regularity theory for the minimizers of integral shape functionals, even in the simplest case
the regularity of the optimal sets was out of reach.
In this paper we prove the first general regularity result for the optimal shapes of integral functionals. In order to make our main result (Theorem 1.2) easier to read, we introduce the following definition.
Definition 1.1
Let D be an open set in \(\mathbb {R}^d\). For \(k\in \mathbb {N}\setminus \{0\}\), \(\alpha \in [0,1]\) and \(N\in \mathbb {N}\), we call a set \(\Omega \subset D\) \((k,\alpha , N)\)-regular in D if the free boundary \(\partial \Omega \cap D\) is the disjoint union of a regular part \(Reg(\partial \Omega )\) and a (possibly empty) singular part \(\text { Sing}(\partial \Omega )\) such that:
-
\(\text { Reg}(\partial \Omega )\) is a relatively open subset of \(\partial \Omega \cap D\) and locally a graph of a \(C^{k,\alpha }\)-regular function;
-
\(\text { Sing}(\partial \Omega )\) is a closed subset of \(\partial \Omega \cap D\) and has the following properties:
-
If \(d<N\), then \(\text { Sing}(\partial \Omega )\) is empty.
-
If \(d\ge N\), then the Hausdorff dimension of \(\text { Sing}(\partial \Omega )\) is at most \(d-N\), namely
$$\begin{aligned} \mathcal {H}^{d-N+{\varepsilon }}\big (\text { Sing}(\partial \Omega )\big )=0\quad \text {for every }{\varepsilon }>0. \end{aligned}$$
-
Moreover, if the regular part \(\text { Reg}(\partial \Omega )\) is \(C^\infty \), then we say that \(\Omega \) is \((\infty , N)\)-regular in D.
The main result of the present paper is the following.
Theorem 1.2
Let D be a bounded open set in \(\mathbb {R}^d\), where \(d\ge 2\). Let
be given non-negative functions. Suppose that the following conditions hold:
-
(a)
\(f,g\in L^\infty (D)\cap C^2(D)\);
-
(b)
there are constants \(C_1,C_2 > 0\) such that
$$\begin{aligned} 0\le C_1 g\le f\le C_2 g\quad \text{ in }\quad D. \end{aligned}$$(1.2) -
(c)
\(Q\in C^{2}(D)\) and there are a positive constants \(c_Q,C_Q\) such that
$$\begin{aligned} 0<c_Q\le Q(x)\leqq C_Q\quad \text {for every}\quad x\in D. \end{aligned}$$
Then, there is \(\alpha \in (0,1)\) such that every solution \(\Omega \subset D\) to the shape optimization problem
is \((1,\alpha ,5)\)-regular in D. Moreover, if \(f,g,Q\in C^\infty (D)\), then \(\Omega \) is \((\infty ,5)\)-regular in D.
Proof
The definitions of the regular and the singular parts, \(\text { Reg}(\partial \Omega )\) and \(\text { Sing}(\partial \Omega )\), of the free boundary \(\partial \Omega \cap D\) are given in Sect. 5. The \(C^{1,\alpha }\) regularity of \(\text { Reg}(\partial \Omega )\) is proved in Theorem 6.4, while the \(C^\infty \) regularity follows from Proposition 6.5. The bounds on the dimension of \(\text { Sing}(\partial \Omega )\) are given in Theorem 8.1. \(\square \)
Remark 1.3
(On the bound on the dimension of the singular set \(\text { Sing}(\partial \Omega )\)) In Sect. 7 we develop a theory about the regularity of the stable global solutions of the one-phase Bernoulli problem (the definition of global stable solution is given in Definition 7.3); we show that there is a critical dimension \(d^*\) (see Definition 7.7), in which a singular global solution appears for the first time (see Theorem 7.8), and we prove that the \(d^*\) can be only 5, 6, or 7 (see Theorem 7.9). In Sect. 8 (Theorem 8.1) we use this results to show the following bounds on the singular part \(\text { Sing}(\partial \Omega )\) of an optimal set \(\Omega \), solution to (1.3):
-
if \(d<d^*\), then \(\text { Sing}(\partial \Omega )\) is empty;
-
if \(d\ge d^*\), then the Hausdorff dimension of \(\text { Sing}(\partial \Omega )\) is at most \(d-d^*\).
In particular, since (by Theorem 7.9) \(d^*\ge 5\), we get that:
-
if \(d<5\), then \(\text { Sing}(\partial \Omega )\) is empty;
-
if \(d\ge 5\), then the Hausdorff dimension of \(\text { Sing}(\partial \Omega )\) is at most \(d-5\).
In other words, the precise statement of Theorem 1.2 is that under the conditions (a)–(b)–(c), the optimal sets are \((1,\alpha ,d^*)\)-regular, where \(d^*\) is the critical dimension from Definition 7.7.
Remark 1.4
(On the assumptions (a)–(b)–(c) in Theorem 1.2) The \(C^2\) regularity assumption in (a) is technical and is related to the use we make of the second order variations of the functional J along vector fields (see Sect. 2). The assumption (b) is used in the proofs of the Lipschitz continuity and the non-degeneracy of the state function \(u_\Omega \); we notice that (b) is automatically satisfied when f and g are both bounded from above and from below by positive constants. The bounds from above and below on the weight Q in (c) are usual in the class of Bernoulli-type free boundary problems; these bounds are necessary for the Lipschitz continuity and the non-degeneracy of \(u_\Omega \) (see Sect. 3), which are essential ingredients for the blow-up analysis in Sect. 4. The \(C^2\) regularity of Q, on the other hand, is used again in the computation of the second variation in Sect. 2 and the passage to the blow-up limit in the proof of Theorem 8.1.
Remark 1.5
(On the existence of optimal sets) In [11, Theorem 1.1] it was proved that if D is a bounded open subset of \(\mathbb {R}^d\) and if f, g, Q satisfy the following conditions:
-
\(f,g\in L^\infty (D)\);
-
there are positive constants \(C_1\le C_2\) such that \(0\le C_1g\le f\le C_2g\) in D;
-
\(Q\in L^\infty (D)\), \(Q\ge 0\) in D,
then, there is an open set \(\Omega \subset D\) solution to the shape optimization problem (1.3).
The presence of the inclusion constraint \(\Omega \subset D\) is essential for the existence theory for general shape optimization problems (see for instance [10] and [11]). In the case of integral functionals with affine cost, as the one in (1.3), the inclusion constraint can be removed. In the next theorem, which we prove in Sect. 9, we show that optimal sets exist in \(\mathbb {R}^d\). Moreover, we prove that the optimal sets in \(\mathbb {R}^d\) are bounded, which implies that they are solutions to (1.3) in some sufficiently large ball \(D:=B_R\), so the regularity of the free boundary in \(\mathbb {R}^d\) is a consequence of Theorem 1.2.
Theorem 1.6
In \(\mathbb {R}^d\), \(d\ge 2\), let \(f,g,Q:\mathbb {R}^d\rightarrow \mathbb {R}\) be non-negative functions.
Suppose that the following conditions hold:
-
(a)
\(f,g\in L^\infty (\mathbb {R}^d)\cap L^1(\mathbb {R}^d)\cap C^2(\mathbb {R}^d)\) and that
$$\begin{aligned} f(x)>0\quad \text{ and }\quad g(x)>0\quad \text{ for } \text{ every }\quad x\in \mathbb {R}^d. \end{aligned}$$ -
(b)
\(Q\in C^{2}(\mathbb {R}^d)\) and there are positive constants \(c_Q,C_Q\) such that
$$\begin{aligned} 0<c_Q\le Q(x)\leqq C_Q\quad \text {for every}\quad x\in \mathbb {R}^d. \end{aligned}$$
Then, there is an open set \(\Omega \subset \mathbb {R}^d\), which is a solution to the shape optimization problem
and every solution \(\Omega \) to (1.4) is bounded and \((1,\alpha ,5)\)-regular in \(\mathbb {R}^d\). Moreover, if \(f,g,Q\in C^\infty (\mathbb {R}^d)\), then \(\Omega \) is \((\infty ,5)\)-regular in \(\mathbb {R}^d\).
1.1 Integral Shape Functionals in the Case \(f=cg\)
In this section we briefly discuss the case in which f and g are proportional, which is the only instance of integral functional studied in the literature. Precisely, we claim that if f and g are such that
the shape optimization problem (1.3) is equivalent to the Bernoulli free boundary problem
Fix a solution \(u\in H^1_0(D)\) to (1.5), for which the set \(\{u\ne 0\}\) is open, and fix an optimal set \(\Omega \) for (1.3) with a state function \(u_\Omega \). Since u satisfies
by integrating by parts, we have that
Analogously, since \(\{u_\Omega \ne 0\}\subset \Omega \) and Q is positive, we have
Thus, if u is a solution to (1.5), then \(J(\{u\ne 0\})\le J(\Omega )\) and so, \(\{u\ne 0\}\) is a solution to (1.3). Conversely, if \(\Omega \) minimizes (1.3), then \(u_\Omega \) is a minimizer of (1.5). Thus, for proportional f and g, the problem (1.3) is equivalent to (1.5); moreover, we notice that the argument above does not require the positivity of f and g, so the equivalence of the problems (1.5) and (1.3) holds also when f and g change sign.
The regularity of the solutions to the free boundary problem (1.5) is nowadays well-understood (at least when Q satisfies the condition (c) of Theorem 1.2). When \(f\ge 0\), the regularity of \(\partial \Omega \) follows from the regularity theory for the one-phase Bernoulli problem (see [2, 14, 18,19,20, 22, 39]). If the right-hand side f changes sign, then (1.5) becomes a two-phase Bernoulli problem, for which the regularity of the free boundary was obtained recently in [35] and [16].
Finally, we notice that when f and g are not proportional, the state function \(u_\Omega \) of an optimal set \(\Omega \) (that is, a solution to the shape optimization problem (1.3)) is not a minimizer of a free boundary functional as the one in (1.5). In particular, this implies that one can test the optimality of \(u_\Omega \) only with functions \({\widetilde{u}}\) which are themselves state functions of some \(\widetilde{\Omega }\). This means that a function \({\widetilde{u}}\), that differs from u only in a small ball \(B_r\), cannot be used to test the optimality of \(u_\Omega \) (truncations, harmonic replacements and radial extensions in small balls are not admitted), which makes most of the classical free boundary regularity results impossible to apply.
1.2 Adjoint State and Optimality Condition on the Free Boundary
Let us go back to the general case when f and g are not proportional. We will show that the optimality condition on the boundary \(\partial \Omega \cap D\) of an optimal open set \(\Omega \) for (1.3) leads to a free boundary problem involving the state function \(u_\Omega \). In order to see this, we introduce the adjoint state function \(v_\Omega \) as follows: for every open set \(A\subset D\) we will denote by \(v_A\) the weak solution to the problem
By an integration by parts one can see that
which means that the two state variables \(u_A\) and \(v_A\) are interchangeable. Precisely, an open set \(\Omega \subset D\) is a solution to (1.3) if and only if it minimizes
Sometimes, it is more convenient to consider simultaneously the two state functions, by using the following equivalent formulation, which is symmetric in \(v_\Omega \) and \(u_\Omega \)
Throughout the paper we will denote the functional from (1.3) by \(\mathcal {F}\). Precisely, we set
which after an integration by parts has the symmetric form
so if \(\xi \in C^\infty _c(D;\mathbb {R}^d)\) is a smooth compactly supported vector field in D and \(\Omega _t:=(Id+t\xi )(\Omega )\), then the first variation of \(\mathcal {F}\) along \(\xi \) is given by (see Lemma 2.6)
Moreover, if \(\Omega \) is a minimizer and \(\partial \Omega \) is smooth, then an integration by parts gives that
so the state functions \(u_\Omega \) and \(v_\Omega \) are (at least formally) solutions to the free boundary system
Remark 1.7
An epsilon-regularity theorem for viscosity solutions of the above system was proved recently in [28]. Precisely, it was shown that if:
-
\(u_\Omega ,v_\Omega :D\rightarrow \mathbb {R}\) are continuous and non-negative functions such that
$$\begin{aligned} \Omega =\{u_\Omega>0\}=\{v_\Omega >0\}; \end{aligned}$$ -
\((\Omega ,u_\Omega ,v_\Omega )\) is a viscosity solutions to (1.8) ;
-
\(u_\Omega \) and \(v_\Omega \) are \({\varepsilon }\)-flat (in a suitable sense) in a ball \(B_r(x_0)\subset D\) centered on \(\partial \Omega \,\);
then the free boundary \(\partial \Omega \) is \(C^{1,\alpha }\)-regular in \(B_{{r}/2}(x_0)\).
We notice that an epsilon-regularity theorem for a similar free boundary system, with \(f\equiv g\equiv 0\), was studied in [1] in the context of a different shape optimization problem. Precisely, in [1], by using the minimality of \(\Omega \), it was shown that it is an NTA domain, so by the Boundary Harnack Principle for harmonic functions, the system reduces to a one-phase problem for the function \(u_\Omega \), for which the epsilon-regularity theorem is known by [2] and [18]. In Sect. A we will detail how our results can be applied to the problem of [1].
Remark 1.8
The free boundary system (1.8) emphasizes that, despite the use of the adjoint variable \(v_\Omega \), it is not possible to reformulate the shape optimization problem (1.3) in terms of an integral functional depending on \(u_\Omega \) or the couple \((u_\Omega ,v_\Omega )\). Precisely, no functional of the form
has Euler-Lagrange equations given by (1.8). The same holds for the equations satisfied by the blow-up limits of \((u_\Omega ,v_\Omega )\). Indeed, since the couple \((u_\Omega ,v_\Omega )\) satisfies (1.8), any 1-homogeneous blow-up limit \((u_0,v_0)\) would be a solution of the same system with \(f=g\equiv 0\) and \(Q\equiv const\,>0\), but this system is not the Euler-Lagrange system of an integral functional of the above type. This peculiarity represents one of the major challenges for studying the regularity of the free boundary and requires rewriting the optimality and stability conditions in terms of inner variations.
1.3 Regularity of the Free Boundary and Dimension of the Singular Set
The key steps in the proof of Theorem 1.2 are the following:
-
Prove that if \(\Omega \) is an optimal set for (1.3), then it is a viscosity solution to (1.8). This, in combination with the epsilon-regularity result cited in Remark 1.7, will imply that the flat part of the free boundary is smooth.
-
Show that there exists a critical dimension \(d^*\in \{5,6,7\}\) such that in dimension \(d< d^*\) all the points on \(\partial \Omega \cap D\) are regular.
-
Prove that a Federer-type dimension reduction principle holds for solutions to (1.3).
There are two main difficulties in following the program outlined in the three points above.
The first difficulty is in the fact that the first order optimality condition
is not leading to a monotonicity formula for \(u_\Omega \) and \(v_\Omega \); in particular, we do not know if the blow-up limits of \(u_\Omega \) and \(v_\Omega \) are in general homogeneous.
The second obstruction comes from the impossibility to make external perturbations of \(u_\Omega \) and \(v_\Omega \) (that is, in light of Remark 1.8 no perturbations of the form \({\widetilde{u}}=u_\Omega +\phi \) and \({\widetilde{v}}=v_\Omega +\psi \) are allowed), so the only information conserved along blow-up sequences is the one contained in the internal variations of \(\Omega \) along smooth vector fields.
We overcome these difficulties by using the first and the second variation of \(\mathcal {F}\). Indeed, suppose that \(\Omega \) is optimal in D and consider the flow \(\Phi _t\) associated to a compactly supported vector field \(\xi \in C^\infty _c(D;\mathbb {R}^d)\). Then, setting \(\Omega _t:=\Phi _t(\Omega )\), we get that the function
has a minumum in \(t=0\), so we have
that is, \(\Omega \) is a stable critical point of the shape functional
We prove that this notion is stable under blow-up limits at free boundary points \(x_0\in \partial \Omega \):
Thus \(\Omega _0=\{u_0>0\}=\{v_0>0\}\) is a stable critical point for the same functional, but this time with \(f\equiv g\equiv 0\) and \(Q\equiv Q(x_0)\). Now, since \(u_0,v_0:\mathbb {R}^d\rightarrow \mathbb {R}\) are harmonic on \(\Omega _0\), we can apply the Boundary Harnack Principle from [29] to obtain that the ratio
is Hölder continuous in \(\Omega _0\), up to \(\partial \Omega _0\). After a second blow-up (this time in zero), we get that the positivity set \(\Omega _{00}\) of the functions
is a stable critical point (in every ball \(B_R\subset \mathbb {R}^d\)) for the functional \(\mathcal {F}(\cdot ,B_R)\), still with \(f\equiv g\equiv 0\) and \(Q\equiv Q(x_0)\). Moreover, by the Boundary Harnack Principle, we also have that \(u_{00}\) and \(v_{00}\) are proportional. Thus, \(u_{00}\) is (up to a constant) a stable critical point (in the sense of (1.9)) for the Alt–Caffarelli’s one-phase functional
so the third blow-up in zero
is a 1-homogeneous stable critical point for the Alt–Caffarelli’s functional \(\mathcal G\).
The existence of a homogeneous blow-up \(u_{000}\) is a key element in the proof of Theorem 1.2. Indeed, it allows to prove (see Proposition 6.3) that the state functions \(u_\Omega \) and \(v_\Omega \) are viscosity solutions to the system (1.8) by showing that if we take a free boundary point admitting a one-sided tangent ball, then the homogeneous blow-up limits \(u_{000}\) and \(v_{000}\) constructed above are half-plane solutions. This, in combination with the epsilon-regularity theorem cited in Remark 1.7, implies that, in a neighborhood of any boundary point admitting a half-plane solution as blow-up limit, the free boundary is \(C^{1,\alpha }\)-regular; we will call these points regular points (see Sect. 5), while the remaining part of the free boundary (if any) will be called singular. Finally, we notice that the smoothness of the regular part requires only the criticality of \(\Omega \) (the first part of (1.9)).
It is natural to expect that the stability of \(\Omega \) (the second part of (1.9)) leads to an estimate on the dimension of the singular set; in fact, the bounds on the critical dimension (the dimension in which a singularity appears for the first time) for minimizers of the one-phase Alt–Caffarelli functional rely (see [14] and [22]) on the well-known stability inequality of Caffarelli-Jerison-Kenig, which was originally obtained in [14] through a particular second order variation of the functional \(\mathcal G\). On the other hand, this stability inequality is not easy to handle when it comes to passing to blow-up limits and developing a dimension reduction principle. In Sect. 7, we give a different formulation of the stability, which uses the second variation along vector fields, as defined in (1.9). Again in Sect. 7, we show that our notion of stability allows to develop a dimension reduction principle and that there is a critical dimension \(d^*\), which is the smallest dimension admitting one-homogeneous stable solutions with singularity (see Theorem 7.8). Then, in Proposition 7.12, we prove that on smooth cones (that is, cones with isolated singularity) our notion of stability is equivalent to the stability inequality of Caffarelli-Jerison-Kenig. Thus, we obtain the bound \(5\le d^*\le 7\) on the critical dimension \(d^*\) as a consequence of the results of Jerison-Savin [22] and De Silva-Jerison [19]. Finally, in Theorem 8.1, we prove the bounds on the singular set from Theorem 1.2 by (again) a triple blow-up argument, which allows to transfer the dimension bounds from Theorem 7.8 to the singular set of the solutions to (1.3).
Remark 1.9
The methodology we developed for the analysis of (1.3) can be applied, with some natural adjustments, to various free boundary and shape optimization problems. In fact, once the local behavior of the state variables close to the free boundary is known, the triple blow-up analysis combined with the notion of stability along domain variations can be applied in an almost straightforward manner. We mention some recent contributions where the authors leveraged our methodology, adapting it to different problems, see [7, 23, 33]. In particular, in [7] the \({\varepsilon }\)-regularity theory was applied to a spectral shape optimization problem, while in [33] and [23] our approach to the stable critical points was applied to free boundary problems.
1.4 Plan of the Paper
In Sect. 2 we compute the first and second variations of the functional \(\mathcal {F}\). In Sect. 3 we prove the non-degeneracy and Lipschitz continuity of the state variables. In Sect. 4 we perform a blow-up analysis by proving the existence of triple blow-up sequences converging to a homogeneous limit.
In Sect. 5, we use the blow-up limits from Sect. 4 to define the decomposition of the free boundary into a regular part and a singular part.
In Sect. 6 we prove that the state functions \(u_\Omega , v_\Omega \) of an optimal set \(\Omega \) for (1.3) are viscosity solutions of the free boundary system (1.8). Using this information, in Theorem 6.4, we prove that the regular part of the free boundary is \(C^{1,\alpha }\)-smooth.
In Sect. 7 we define the notion of a global stable solution to the one-phase Bernoulli (Alt–Caffarelli) problem and we study the dimension of the singular sets for these global stable solutions. We notice that this section can also be read indipendently and that, together with Sect. 2, it contains the key results for the analysis of the singular set.
In Sect. 8 we use the first and the second variations from Sect. 2, the triple blow-up procedure from Sect. 4 and the theory from Sect. 7 in order to prove the dimension bounds on the singular set. This concludes the proof of Theorem 1.2, which follows from Theorem 8.1, Theorem 6.4 and Proposition 6.5.
In Sect. 9 we address the existence of optimal sets in \(\mathbb {R}^d\) and we prove Theorem 1.6. Ultimately, in Appendix A we apply the analysis of Sects. 7 and 8 to optimal sets arising sets arising from the heat conduction problem studied in [1].
1.5 Notations
In the whole paper we use the notation
for the ball of radius r centered at point \(x_0\) and, when \(x_0=0\), we write \(B_r=B_r(0)\); we denote by \(\omega _d\) the Lebesgue measure of a ball of radius one in \(\mathbb {R}^d\). For any set \(A\subset \mathbb {R}^d\), we set
Given a non-negative function u we often denote its positivity set as \(\Omega _u:=\{u>0\}\).
We denote by \(H^1(\mathbb {R}^d)\) the set of Sobolev functions in \(\mathbb {R}^d\), that is, the closure of the smooth functions with bounded support \(C^\infty _c(\mathbb {R}^d)\) with respect to the usual Sobolev norm
Given an open set \(\Omega \subset \mathbb {R}^d\), we define the Sobolev space \(H^1_0(\Omega )\) as the closure, with respect to \(\Vert \cdot \Vert _{H^1}\), of the space \(C^\infty _c(\Omega )\) of smooth functions compactly supported in \(\Omega \). Thus, every \(u\in H^1_0(\Omega )\) is identically zero outside \(\Omega \) and we have the inclusion \(H^1_0(\Omega )\subset H^1(\mathbb {R}^d)\).
We will sometimes use the following notation for minimum and maximum of two real numbers:
Given a function \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) we will denote by \(\nabla u\) and Du the vectors column and row with components the partial derivatives of u, while \(D^2u\) will be the Hessian matrix of u. Given a vector field \(F:\mathbb {R}^d\rightarrow \mathbb {R}^d\) with components \(F_k\), \(k=1,\dots ,d\) we will denote by \(\nabla F\) the \(d\times d\) matrix with columns \(\nabla F_k\), \(k=1,\dots , d\) and rows \((\partial _jF_1,\partial _jF_2,\dots ,\partial _jF_d)\). By convention \(DF:=(\nabla F)^T\), where for any matrix \(M\in \mathbb {R}^{d\times d}\), we will denote by \(M^T\) its transpose. Given a vector field \(V:\mathbb {R}^d\rightarrow \mathbb {R}^d\) and a matrix with variable coefficients \(M=(m_{ij})_{ij}:\mathbb {R}^d\rightarrow \mathbb {R}^{d\times d}\) we will denote by \((V\cdot \nabla )(M)\) the \(d\times d\) matrix with variable coefficients \(V\cdot \nabla m_{ij}\).
2 First and Second Variations Under Inner Perturbations
In this section we compute the first and the second variations (with respect to perturbations with compactly supported vector fields) of the functional \(\mathcal {F}\) from (1.7). Both variations will be fundamental tools in the study of the blow-up limits of the state variables \(u_\Omega \) and \(v_\Omega \) on a domain \(\Omega \), which is optimal for (1.3).
2.1 First and Second Variation of the State Function
In Lemma 2.5 we compute the expansion of the state variable \(u_\Omega \) (solution to (1.1)) with respect to smooth perturbations of a set \(\Omega \). We first prove Lemma 2.2 and Lemma 2.3, where we compute the expansion of a one-parameter family of solutions to PDEs on the same domain \(\Omega \).
Remark 2.1
In what follows we will denote by \(\mathbb {R}^{d\times d}\) the space of \(d\times d\) square matrices with real coefficients. Given a real matrix \(A=(a_{ij})_{ij}\in \mathbb {R}^{d\times d}\), we define its norm in the space \(\mathbb {R}^{d\times d}\) as
and we notice that for every vector \(V\in \mathbb {R}^d\), we have \(|AV|\le \Vert A\Vert _{\mathbb {R}^{d\times d}}|V|\), where |V| is the usual Euclidean norm of V. Next, let \(\Omega \) be a measurable set in \(\mathbb {R}^d\). Given a matrix \(A:\Omega \rightarrow \mathbb {R}^{d\times d}\) with variable coefficients \(a_{ij}:\Omega \rightarrow \mathbb {R}\), we say that
if \(a_{ij}\in L^\infty (\Omega )\) for every \(1\le i,j\le d\). We define the norm \(\Vert \cdot \Vert _{L^\infty (\Omega ;\mathbb {R}^{d\times d})}\) as
Lemma 2.2
(First order expansion of solutions to PDEs) Let \(\Omega \) be a bounded open set in \(\mathbb {R}^d\). Let the functions
be such that:
-
(a)
\(A_t(x)\) is a symmetric matrix for every \((t,x)\in \mathbb {R}\times \Omega \) and there is a symmetric matrix \(\delta A\in L^\infty (\Omega ;\mathbb {R}^{d\times d})\) such that
$$\begin{aligned} A_t=\text { Id}+t(\delta A)+o(t)\quad \text {in }L^\infty (\Omega ;\mathbb {R}^{d\times d}). \end{aligned}$$ -
(b)
there is \(\delta f\in L^2(\Omega )\) such that
$$\begin{aligned} f_t=f_0+t(\delta f)+o(t)\quad \text {in }L^2(\Omega ). \end{aligned}$$
Then, for every t small enough there is a unique solution \(u_t\) to the problem
and
where \(\delta u\) is the unique weak solution in \(H^1_0(\Omega )\) to the PDE
Proof
Clearly \(u_0\in H^1_0(\Omega )\) is the solution to \(-\Delta u_0=f_0\) in \(\Omega \). We set \(w_t:=\frac{1}{t}(u_t-u_0)\). We will prove that \(w_t\) converges to \(\delta u\) strongly in \(H^1_0(\Omega )\).
We notice that (2.1) can be written as
So, using the equation for \(u_0\) and dividing by t, we get
If we fix \({\varepsilon }>0\), we can choose t small enough such that
Thus, by testing (2.3) with \(w_t\), we obtain
Now, by the Poincaré inequality,
and the equation for \(u_0\), we have that
which gives the bound
Thus, we deduce that
and so, for t (and \({\varepsilon }<1\)) small enough
Thus, for every sequence \(t_n\rightarrow 0\), there is a subsequence for which \(w_{t_n}\) converges as \(n\rightarrow \infty \), strongly in \(L^2(\Omega )\) and weakly in \(H^1_0(\Omega )\), to some function \(w_\infty \). Passing to the limit the Eq. (2.3) we get that \(w_\infty \) is also a solution to (2.2). Thus \(w_\infty =\delta u\). In particular, this implies that \(w_t\) converges as \(t\rightarrow 0\), strongly in \(L^2(\Omega )\) and weakly in \(H^1_0(\Omega )\), to \(\delta u\). Finally, in order to prove that the convergence is strong, we test again (2.3) with \(w_t\):
Combining this estimate with the lower semi-continuity of the \(H^1\) norm, we get
which implies that \(w_t\) converges to \(\delta u\) strongly in \(H^1_0(\Omega )\). \(\square \)
Lemma 2.3
(Second order expansion of solutions to PDEs) Let \(\Omega \) be a bounded open set in \(\mathbb {R}^d\). Let the functions
be such that:
-
(a)
\(A_t(x)\) is a symmetric matrix for every \((t,x)\in \mathbb {R}\times \Omega \) and there are symmetric matrices \(\delta A\in L^\infty (\Omega ;\mathbb {R}^{d\times d})\) and \(\delta ^2A\in L^\infty (\Omega ;\mathbb {R}^{d\times d})\) such that
$$\begin{aligned} A_t=\text { Id}+t\,\delta A+t^2\delta ^2A+o(t^2)\quad \text {in }L^\infty (\Omega ;\mathbb {R}^{d\times d}); \end{aligned}$$ -
(b)
there are \(\delta f\in L^2(\Omega )\) and \(\delta ^2f\in L^2(\Omega )\) such that
$$\begin{aligned} f_t=f_0+t\,\delta f+t^2\delta ^2f+o(t^2)\quad \text {in }L^2(\Omega ). \end{aligned}$$
Then, for every t small enough there is a unique solution \(u_t\) to the problem
and
where \(\delta u\in H^1_0(\Omega )\) is the solution to (2.2) and where \(\delta ^2u\in H^1_0(\Omega )\) solves the PDE
Proof
Let \(w_t:=\frac{1}{t}(u_t-u_0)\) be as in the proof of Lemma 2.2. We set
We will prove that \(v_t\) converges strongly in \(H^1_0(\Omega )\) to \(\delta ^2u\). From the equation for \(w_t\), we have
Thus, using the Eq. (2.2) for \(\delta u\) (\(-\Delta (\delta u)-{{\,\textrm{div}\,}}(\delta A\nabla u_0)=\delta f\)), we get
Now, reasoning as in Lemma 2.2, we get that the family \(v_t\) is uniformly bounded in \(H^1_0(\Omega )\) and converges as \(t\rightarrow 0\) strongly in \(L^2(\Omega )\) and weakly in \(H^1_0(\Omega )\) to the solution \(\delta ^2u\) of (2.4). In order to obtain the strong \(H^1\) convergence, we compute
which concludes the proof. \(\square \)
Remark 2.4
We recall that if \(M=(m_{ij})_{1\le i,j\le d}\in \mathbb {R}^{d\times d}\) is a matrix with constant coefficients, then the following Taylor expansions in \(\mathbb {R}^{d\times d}\) (with respect to the norm \(\Vert \cdot \Vert _{\mathbb {R}^{d\times d}}\))
where \(\text { Id}\) is the identity matrix in \(\mathbb {R}^{d\times d}\) and \(\text { tr}(M)\) is the trace \(\text { tr}(M):=\sum _{i=1}^dm_{ii}\). As a consequence, if \(M\in L^\infty (\Omega ;\mathbb {R}^{d\times d})\) is a matrix with variable coefficients, then we have the expansions
In particular, this implies that given two matrices \(M,N\in L^\infty (\Omega ;\mathbb {R}^{d\times d})\), we have:
and so,
Proposition 2.5
Let D be a bounded open set in \(\mathbb {R}^d\) and let \(f \in C^2(D)\) with \(f\ge 0\) in D. Given an open set \(\Omega \subset D\) and a compactly supported vector field \(\xi \in C^\infty _c(D;\mathbb {R}^d)\), we consider the associated flow \(\Phi :\mathbb {R}\times \mathbb {R}^d\rightarrow \mathbb {R}^d\), determined by the family of ODEs (for every \(x\in D\))
We consider the family of open sets \(\Omega _t:= \Phi _t(\Omega )\) and the corresponding state variables \(u_{\Omega _t}\) given by (1.1). We define \(\delta u_\Omega \) and \(\delta ^2 u_\Omega \) to be the weak solutions in \(H^1_0(\Omega )\) to the PDEs
and
where the matrices \(\delta A\in L^\infty (D;\mathbb {R}^{d\times d})\) and \(\delta ^2A\in L^\infty (D;\mathbb {R}^{d\times d})\) are given by
while the variations \(\delta f\in L^2(D)\) and \(\delta ^2f\in L^2(D)\) of the right-hand side f are:
Then,
Proof
We set \(u_t:=u_{\Omega _t}\circ \Phi _t\). Then, \(u_t\in H^1_0(\Omega )\) and \(u_{\Omega _t}=u_t\circ \Phi _t^{-1}\).
Moreover, by a change of variables, we have that \(u_t\) is satisfies the PDE
where the matrix \(A_t\) and the function \(f_t\) are defined as
where for a vector field \(F:\mathbb {R}^d\rightarrow \mathbb {R}^d\) with components \(F_k\), \(k=1,\dots ,d\), we denote by
DF the matrix with rows \(DF_k=(\nabla F_k)^T\). We next compute the second order Taylor expansion of \(D\Phi _t\) in \(t=0\). By differentiating the equation for the flow \(\Phi _t\), we get
Then, taking another derivative in t, we get
where for a vector field F and a matrix \(M=(m_{ij})_{ij}\), we use the notation \((F\cdot \nabla )[M]\) for the matrix with coefficients \(F\cdot \nabla m_{ij}\). Finally, taking \(t=0\), we get
and the Taylor expansion
By the expansions from Remark 2.4, we get
where \(\delta A\), \(\delta ^2A\), \(\delta f\), \(\delta ^2f\) are given by (2.6) and (2.7). Thus, the claim follows from Lemmas 2.2 and 2.3. \(\square \)
2.2 First and Second Variation of \(\mathcal {F}\)
In the next lemma, we compute the first derivative of the functional \(\mathcal {F}\) along inner variations with compact support in D.
Lemma 2.6
(First variation of \(\mathcal {F}\) along inner perturbations) Let D be a bounded open set in \(\mathbb {R}^d\) and let \(f,g,Q\in C^1(D)\). Let \(\Omega \subset D\) be open and \(\xi \in C^\infty _c(D;\mathbb {R}^d)\) be a vector field with compact support in D. Let \(\Phi _t\) be the flow of the vector field \(\xi \) defined by (2.5) and set \(\Omega _t:= \Phi _t(\Omega )\). Then
Moreover, if \(\partial \Omega \) is \(C^2\)-regular in a neighborhood of the support of \(\xi \), then
where \(\nu \) is the outer unit normal to \(\partial \Omega \).
Proof
By applying Lemma 2.5 to \(u_t=u_{\Omega _t}\circ \Phi _t\) and \(v_t=v_{\Omega _t}\circ \Phi _t,\) we get that
where \(\delta u_\Omega \) and \(\delta v_\Omega \) are the solutions to
with \(\delta f:={{\,\textrm{div}\,}}(f\xi )\), \(\delta g:={{\,\textrm{div}\,}}(g\xi )\), and \(\delta A\) as in Lemma 2.5. Therefore, setting
we get
where in the first equality we applied the change of variables \(y=\Phi _t(x)\) and in the last one we use the equations \(-\Delta u_\Omega = f\) and \(-\Delta v_\Omega =g\) in \(\Omega \). Substituting with the expression for \(\delta A\) from (2.6), we obtain (2.8).
Suppose now that \(\partial \Omega \) is \(C^2\)-smooth. Since in \(\Omega \) we have the identity
by integrating by parts we get
Since \(u_\Omega \) and \(v_\Omega \) are positive in \(\Omega \) and vanish on \(\partial \Omega \), we have that
and so
which concludes the proof. \(\square \)
Remark 2.7
(First variation and stationary domains) Given a bounded open set D, functions f, g and Q on D, and the functional \(\mathcal {F}\) defined in (1.7), we will use the notation \(\delta \mathcal {F}(\Omega ,D)[\xi ]\) for the first variation of \(\mathcal {F}\) at \(\Omega \) along a smooth compactly suppported vector field \(\xi \in C^\infty _c(D;\mathbb {R}^d)\). Precisely, we set
We will say that an open set \(\Omega \subset D\) is stationary (or a critical point) for \(\mathcal {F}\) in D if
By (2.9), if \(\Omega \) is stationary in D and the boundary \(\partial \Omega \cap D\) is smooth, then
Finally, we notice that, by Lemma 2.5, any minimizer of (1.3) is stationary for \(\mathcal {F}\).
Proposition 2.8
(Second variation of \(\mathcal {F}\) along inner perturbations) Let \(D\subset \mathbb {R}^d\) be a bounded open set in \(\mathbb {R}^d\) and let \(f,g,Q\in C^2(D)\). Let \(\Omega \subset D\) be an open set and \(\xi \in C^\infty _c(D;\mathbb {R}^d)\) be a smooth vector field with compact support. Let \(\Phi _t\) be the flow of the vector field \(\xi \) defined by (2.5) and set \(\Omega _t:= \Phi _t(\Omega )\). Then
where \(\delta ^2 A,\delta ^2 f\), \(\delta ^2 g\), \(\delta u_\Omega \) and \(\delta v_\Omega \) are the ones defined in Lemma 2.5 and where
Proof
By applying Lemma 2.5 to \(u_t:=u_{\Omega _t}\circ \Phi _t\) and \(v_t:=v_{\Omega _t}\circ \Phi _t\), we get that
with \(\delta u_\Omega ,\delta v_\Omega \) satisfying (2.10) and \(\delta ^2 u_\Omega ,\delta ^2 v_\Omega \in H^1_0(\Omega )\) such that
Thus, by computing the second order (in t) Taylor expansion of
we get that
Thus, we only have to show that we can write the above expression as in (2.12). By using \(\delta v_\Omega \) as a test function in the equation for \(\delta u_\Omega \) and vice versa, we obtain
Then, by testing the equations for \(u_\Omega \) and \(v_\Omega \) respectively with \(\delta ^2v_\Omega \) and \(\delta ^2u_\Omega \), we get
Using this identities in the expression of the second derivative, we get precisely (2.12). \(\square \)
Remark 2.9
(Second variation and stable critical domains) Given a bounded open set D, functions f, g and Q on D, and the functional \(\mathcal {F}\) from (1.7), we will indicate by \(\delta ^2 \mathcal {F}(\Omega ,D)[\xi ]\) the second variation of \(\mathcal {F}\) at \(\Omega \) along a smooth compactly supported vector field \(\xi \in C^\infty _c(D;\mathbb {R}^d)\). Precisely, we set
In particular, we notice that for every open set \(\Omega \), we have
where \(\Omega _t=\Phi _t(\Omega )\) (with \(\Phi _t\) the flow associated to the vector field \(\xi \)) and \(\delta \mathcal {F}(\Omega ,D)[\xi ]\) is the first variation (2.11).
We will say that an open set \(\Omega \subset D\) is a stable critical point for \(\mathcal {F}\) in D if
By Lemmas 2.5 and 2.8, any minimizer of (1.3) is a stable critical point for \(\mathcal {F}\).
3 Lipschitz Regularity and Non-degeneracy of the State Functions
In this section we study the regularity of the state functions \(u_\Omega \) and \(v_\Omega \) on an optimal domain \(\Omega \), as well as their behavior close to the free boundary \(\partial \Omega \). Consequently, we prove that the set \(\Omega \) satisfies some density estimates.
3.1 Assumptions on D, f, g, and Q
Throughout this subsection we will assume that:
-
D is a open subset of \(\mathbb {R}^d\), with \(d\ge 2\);
-
\(f,g\in L^\infty (D)\) are two functions such that
$$\begin{aligned} \Vert f\Vert _{L^\infty }+\Vert g\Vert _{L^\infty }\le M\quad \text {and}\quad 0\le C_1g\le f\le C_2g\quad \text {on }D, \end{aligned}$$for some positive constants \(M,C_1,C_2>0\).
-
\(Q\in L^\infty (D)\) is such that
$$\begin{aligned} 0<c_Q\le Q\le C_Q\quad \text {on}\quad D. \end{aligned}$$
3.2 Inwards and Outwards Minimality Conditions
We will use the following notation. Given a set \(A\subset \mathbb {R}^d\), and functions \(f\in L^2(A)\) and \(\varphi \in H^1(A)\), we set
Proposition 3.1
Let \(D\subset \mathbb {R}^d\) and \(f,g,Q\in L^\infty (D)\) be as in Sect. 3.1 and let \(\Omega \subset D\) be an open set that minimizes (1.3) in D. Then, the solution \(u_\Omega \) to (1.1) has the following properties.
-
(i)
Outwards minimality. For every open set \(\widetilde{\Omega }\subset D\) such that \(\Omega \subset \widetilde{\Omega }\) we have
$$\begin{aligned} E_f(u,D) + \frac{C_2C_Q}{2}|\Omega |\le E_f(\phi ,D)+\frac{C_2C_Q}{2}|\widetilde{\Omega }|\qquad \text{ for } \text{ every } \phi \in H^1_0(\widetilde{\Omega }). \end{aligned}$$In particular, for every \(B_r(x_0)\subset D\), we have
$$\begin{aligned} E_f(u,B_r(x_0))\le E_f(\phi ,B_r(x_0)) + \frac{C_2C_Q}{2}\omega _d r^d, \end{aligned}$$(3.1)for every \(\phi \in H^1(B_r(x_0))\) such that \(\phi -u_\Omega \in H^1_0(B_r(x_0))\).
-
(ii)
Inwards minimality. For every open set \(\omega \subset \Omega \) we have
$$\begin{aligned} E_f(u,D) + \frac{C_1c_Q}{2}|\Omega |\le E_f(\phi ,D) + \frac{C_1c_Q}{2}|\omega |\qquad \text{ for } \text{ every } \phi \in H^1_0(\omega ).\nonumber \\ \end{aligned}$$(3.2)
Proof
Suppose that the open set \(\widetilde{\Omega }\subset D\) contains \(\Omega \). Then, the optimality of \(\Omega \) implies that
which can be written as
Now, the positivity of f implies that \(u_{\widetilde{\Omega }}\ge u_\Omega \) on D, so we get
which after rearranging the terms gives
which after an integration by parts on \(\widetilde{\Omega }\) and \(\Omega \), reads as
Finally, since \(u_{\widetilde{\Omega }}\) minimizes the energy \(E_f(\cdot ,D )\) among all functions in \(H^1_0(\widetilde{\Omega })\), we obtain (i).
The proof of (ii) is similar. Let \(\omega \subset \Omega \) and let \(u_\omega \) be the associated state function. Then, using the optimality of \(\Omega \) and the bounds on f, g and Q, we get
which implies (ii) since \(u_\omega \) minimizes \(E_f(\cdot ,D )\) in \(H^1_0(\omega )\). \(\square \)
Remark 3.2
The state variable \(v_\Omega \) satisfies analogous inwards/outwards minimality conditions for the functional \(E_g(\cdot ,D)\), where the constants \(C_1\) and \(C_2\) are replaced by \(1/C_2\) and \( 1/C_1\).
3.3 Lipschitz Continuity and Non-degeneracy
As a consequence of Proposition 3.1 we obtain the Lipschitz continuity and the non-degeneracy of \(u_\Omega \) (and of \(v_\Omega \)).
Corollary 3.3
Let \(D\subset \mathbb {R}^d\) and \(f,g,Q\in L^\infty (D)\) be as in Sect. 3.1. If \(\Omega \subset D\) is optimal for (1.3), then the state functions \(u_\Omega \) and \(v_\Omega \) are locally Lipschitz in D with Lipschitz constants depending on d, \(C_1\), \(C_2\), M and \(C_Q\).
Proof
By Proposition 3.1 (i), \(u_\Omega \) satisfies (3.1) for any \(B_r(x_0)\subset D\), so \(u_\Omega \) is an almost-minimizer in the sense of [8, Definition 3.1]. Thus, by [8, Theorem 3.3], \(u_\Omega \) is locally Lipschitz continuous in D with Lipschitz constant depending on d and the bounds from above on \(\left\| {f} \right\| _{L^\infty }\) and \(C_2C_Q\). \(\square \)
Lemma 3.4
Let \(D\subset \mathbb {R}^d\) and \(f,g,Q\in L^\infty (D)\) be as in Sect. 3.1 and let \(\Omega \subset D\) be an optimal set for (1.3) and \(u_\Omega \) be the associated state function. Then, there are constants \(C_0, r_0>0\), depending on d, \(C_1\), \(c_Q\) and M, such that the following implication holds
for every \(x_0 \in \overline{\Omega }\cap D\) and every \(r \in (0,r_0]\). In other words, if \(x_0\in {\overline{\Omega }}\) and \(B_r(x_0)\subset D\), then
Proof
By Proposition 3.1 (ii), we have that for every \(\omega \subset \Omega \)
Thus, the claim follows by [2, Lemma 4.4], [9, Lemma 3.3], or [20, Lemma 2.8]. \(\square \)
3.4 Density Estimates on the Boundary of \(\Omega \)
An a consequence of the Lipschitz continuity and the non-degeneracy of \(u_\Omega \) and \(v_\Omega \), we obtain density estimates for the optimal set \(\Omega \).
Proposition 3.5
Let \(D\subset \mathbb {R}^d\) and \(f,g,Q\in L^\infty (D)\) be as in Sect. 3.1. Then, there are \({\varepsilon }_0, r_0>0\) (depending on \(C_1\), \(C_2\), M, d, \(c_Q\), \(C_Q\)) such that for every set \(\Omega \subset D\) optimal for (1.3)
for every ball \(B_r(x_0)\subset D\) of radius \(r\le r_0\) centered on \(\partial \Omega \).
Proof
Assume that \(x_0=0\in \partial \Omega \). The lower estimate is an immediate consequence of the Lipschitz continuity (Lemma 3.3) and the non-degeneracy (Lemma 3.3) of \(u_\Omega \). The upper bound can be obtained as in [2]. Precisely, consider the solution h to
Since \(\Delta u_\Omega +f\ge 0\) in \(\mathbb {R}^d\), we get that \(-\Delta (h-u_\Omega )\ge M-f\geqq 0\) in \(B_r\). In particular, we have that \(u_\Omega \leqq h\) and \(\{u_\Omega>0\}\subset \{h>0\}\) in \(B_r\). Thus, testing the optimality (3.1) of \(u_\Omega \) with h,
By the Poincaré and Cauchy-Schwarz inequalities, we have
so in order to prove the upper bound in (3.3), we only need to show that \(\frac{1}{r^{d+1}} \int _{B_{r}}(h-u_\Omega )\,dx\) is bounded from below by a positive constant. Notice that, by the non-degeneracy of \(u_\Omega \), we have
On the other hand, since \(h(x)+\frac{M}{2d}|x|^2\) is harmonic in \(B_R\), the Harnack inequality in \(B_r\) implies
Thus, by taking \(r_0\) such that \(2C_dr_0M\le {\widetilde{C}}\), we get that \(h \geqq C_d\widetilde{C} r = \overline{C}r\) in \(B_{{r}/2}.\) On the other hand, if L is the Lipschitz constant of \(u_\Omega \), then for any \({\varepsilon }\in (0,1)\), \(u_\Omega \leqq L {\varepsilon }r\) in \(B_{{\varepsilon }r}\). Then
which concludes the proof after choosing \({\varepsilon }\le 1/2\) small enough. \(\square \)
3.5 An Estimate on the Level Sets of \(u_\Omega \)
We conclude the section with this auxiliary result that will play a crucial role in Sect. 4.3 in the proof of the existence of homogeneous blow-up limits.
Lemma 3.6
Let \(D\subset \mathbb {R}^d\) and \(f,g,Q\in L^\infty (D)\) be as in Sect. 3.1; let \(\Omega \) be a solution to (1.3) and let \(u_\Omega \) be the associated state function. Then, there are constants \(C>0\) and \(r_0>0\), depending only on \(d,M,C_1,C_2,c_Q,C_Q\), such that
for every \(B_r(x_0)\subset D\) centered on \(\partial \Omega \) and every \(t\in (0,1)\).
Proof
The estimate is contained in the proof of [11, Theorem 1.10]; we sketch the idea for the sake of completeness. Let \(x_0 = 0 \in \partial \Omega \) and \(t>0\). We fix a function \(\eta \in C^\infty _c(B_{2r})\) such that \(0\le \eta \le 1\) in \(B_{2r}\) and \(\eta \equiv 1\) in \(B_r\), and we use the competitor
to test the optimality condition (3.2) in \(B_{2r}\). Since
we get
Thus (3.2) implies
Therefore, we get
with \(C>0\) depending on d, \(\Vert u_\Omega \Vert _{L^\infty (B_{2r})}\), M, \(C_1\) and \(c_Q\). \(\square \)
4 Compactness and Convergence of Blow-Up Sequences
Take an optimal set \(\Omega \) for (1.3) in some \(D\subset \mathbb {R}^d\), and consider the corresponding state functions \(u=u_\Omega \) and \(v=v_\Omega \). For any \(x_0 \in \partial \Omega \cap D\) and any sequence \(r_k \rightarrow 0^+\), we set
Since u and v vanish in \(x_0\) and are Lipschitz (in a neighborhood of \(x_0\)), we have that \(u_{x_0,r_k}\) and \(v_{x_0,r_k}\) vanish in 0 and (for large k) are uniformly Lipschitz in any ball \(B_R\). Thus, there are functions \(u_0,v_0:\mathbb {R}^d\rightarrow \mathbb {R}\) and subsequences of \(u_{x_0,r_{k}}\) and \(v_{x_0,r_{k}}\) that converge locally uniformly in \(\mathbb {R}^d\) respectively to \(u_0\) and \(v_0\). As usual we say that \(u_0\) and \(v_0\) are blow-up limits of u and v in \(x_0\); and we recall that they might depend on the sequence \(r_{k}\). We notice that the blow-up limits of u and v will always be taken along the same sequence \(r_k\rightarrow 0\).
The main results are Propositions 4.3 and 4.7. In Proposition 4.3 we list the properties of any couple of functions \(u_0,v_0\) obtained as blow-up limits of u, v, while in Proposition 4.7 we show that there is at least one sequence \(r_k\rightarrow 0\) that provides blow-up limits which are 1-homogeneous and stationary for the one-phase Alt–Caffarelli functional.
4.1 A General Lemma About the Convergence of Blow-Up Sequences
The construction of the blow-up limit from Proposition 4.7 will require taking three consecutive blow-ups. We give here a general lemma, which we will use several times in this section.
Lemma 4.1
Let \(B_{2R}\) be a ball in \(\mathbb {R}^d\). Let \(u_n:{\overline{B}}_{2R}\rightarrow \mathbb {R}\) be a sequence of non-negative Lipschitz functions converging uniformly to a Lipschitz function \(u_\infty :{\overline{B}}_{2R}\rightarrow \mathbb {R}\), and suppose that there is a constant \(L>0\) such that
Then, the following holds.
-
(i)
Suppose that there is a constant \({\tilde{C}}>0\) such that, for every \(n\ge 1\),
$$\begin{aligned} \sup _{B_r(x_0)} u_n \ge \tilde{C} r\quad \text {for every}\quad x_0\in B_R\cap \overline{\{u_n>0\}}\quad \text {and every}\quad r\in (0,R).\nonumber \\ \end{aligned}$$(4.1)Then
$$\begin{aligned} \sup _{B_r(x_0)} u_\infty \ge \tilde{C} r\quad \text {for every}\quad x_0\in B_R\cap \overline{\{u_\infty >0\}}\quad \text {and every}\quad r\in (0,R),\nonumber \\ \end{aligned}$$(4.2)and
$$\begin{aligned} \mathbbm {1}_{\{u_n>0\}}\rightarrow \mathbbm {1}_{\{u_\infty >0\}}\quad \text {pointwise a.e. in }B_R. \end{aligned}$$(4.3) -
(ii)
Suppose that there is a constant \({\tilde{M}}>0\) such that, for every \(n\ge 1\), we have the bound
$$\begin{aligned} \left| \int _{B_{2R}}\nabla u_n\cdot \nabla \varphi \,dx\right| \le \tilde{M} \Vert \varphi \Vert _{L^\infty (B_{2R})}\quad \text {for every }\varphi \in C^{0,1}_c(B_{2R}), \end{aligned}$$(4.4)where \(C^{0,1}_c(B_{2R})\) is the space of Lipschitz functions with compact support in \(B_{2R}\). Then \(u_n\) converges to \(u_\infty \) strongly in \(H^1(B_R)\).
Proof
We first prove (i). Suppose that \(x_0\in B_R\cap \overline{\{u_\infty >0\}}\). Then, there is a sequence \(x_n\rightarrow x_0\) of points \(x_n\in B_R\cap \overline{\{u_\infty >0\}}\). Let \(r_n:=r-|x_n-x_0|\). By (4.1), there is a point \(y_n\in B_{r_n}(x_n)\subset B_r(x_0)\) such that \(u_n(y_n)\ge {\widetilde{C}}r_n\). Thus, by the uniform convergence of \(u_n\), we get (4.2). In order to prove (4.3), we first notice that by the pointwise convergence of \(u_n\) to \(u_\infty \), we have that
so it is sufficient to prove that the set
is of measure zero. Now, by (4.1), we get that for every \(r\in (0,R)\) there is a sequence \(y_n\in {\overline{B}}_r(x_0)\) such that \(u_n(y_n)\ge \tilde{C}r\). Then, by the uniform convergence of \(u_n\), we have that there is \(y_\infty \in {\overline{B}}_r(x_0)\) such that \(u_\infty (y_\infty )\ge \tilde{C}r\). Then, by the Lipschitz continuity of \(u_\infty \), the ball \(B_{\tilde{C} r/L}(y_\infty )\) is contained in \(\{u_\infty >0\}\). Since r is arbitrary, we get that the Lebesgue density of \(\{u_\infty >0\}\) in \(x_0\) cannot be zero, so the set S has zero measure.
We next prove (ii). Since \(u_n\) is uniformly bounded in \(H^1(B_{2R})\), we have that \(u_n\) converges to \(u_\infty \) weakly in \(H^1(B_{2R})\). Thus, the estimate (4.4) holds also for \(u_\infty \), that is
Choose a function \(\varphi \in C^\infty _c(B_{2R})\) such that \(\varphi \equiv 1\) in \(B_R\). Then,
Since the right-hand side converges to zero, we get the claim. \(\square \)
Finally, we notice that the above lemma can be applied to the state functions \(u_\Omega \) and \(v_\Omega \) of an optimal domain. This is a consequence of the following lemma.
Lemma 4.2
Let \(B_{2R}\subset \mathbb {R}^d\) and \(u\in H^1(B_{2R})\) be a non-negative function with the following properties.
-
(a)
There is a function \(f\in L^\infty (B_{2R})\) such that
$$\begin{aligned} -\Delta u=f\quad \text {in }\Omega _u:=\{u>0\},\qquad u=0\quad \text {on }\partial \Omega _u\cap B_{2R}, \end{aligned}$$in the sense that
$$\begin{aligned} \int _{B_{2R}}\nabla u\cdot \nabla \varphi \,dx=\int _{B_{2R}}\varphi f\,dx\quad \text {for every }\varphi \in H^1_0(B_{2R})\quad \text {with}\quad \varphi =0\quad \text {in }B_{2R}\setminus \Omega _u. \end{aligned}$$ -
(b)
There is \(\Lambda >0\) such that, for every non-negative \(\varphi \in H^1_0(B_{2R})\),
$$\begin{aligned} E_f(u,B_{2R})+\frac{\Lambda }{2}|B_{2R}\cap \{u>0\}|\le E_f(u+\varphi ,B_{2R})+\frac{\Lambda }{2}|B_{2R}\cap \{u+\varphi >0\}|. \end{aligned}$$
Then, for every \(\varphi \in C^{0,1}_c(B_{R})\), we have
Proof
We only sketch the proof and we refer to [38, Chapter 3] for the details. By (a) we have that
is a positive Radon measure on \(B_{2R}\), where
By testing the optimality condition in (b) with a function \(\varphi \in C^{0,1}_c(B_{2R})\), we get
and choosing \(\varphi =R\phi \) with \(\phi \equiv 1\) in \(B_{R}\), we obtain
As a consequence, for every \(\varphi \in C^{0,1}_c(B_R)\), we have
which concludes the proof. \(\square \)
4.2 First Blow-Up
In this subsection we list the properties of any couple of blow-ups
of the state functions \(u_\Omega \) and \(v_\Omega \) at a boundary point \(x_0\in \partial \Omega \cap D\). The qualitative properties (Lipschitz continuity, non-degeneracy, density estimates) of \(u_\Omega \) and \(v_\Omega \) are conserved under blow-up limits; the stationarity condition also passes to the limit; the main difference is that \(u_0\) and \(v_0\) are harmonic where they are positive.
Proposition 4.3
Let \(D\subset \mathbb {R}^d\) be a bounded open set, let \(\Omega \) be a solution to (1.3) with f, g, Q as in Theorem 1.2, and let \(u:=u_\Omega \) and \(v:=v_\Omega \) be the state functions on \(\Omega \) defined in (1.1) and (1.6). We consider a point \(x_0\in \partial \Omega \cap D\) and blow-up sequences \(u_{x_0,r_k},v_{x_0,r_k}\) of u, v converging locally uniformly in \(\mathbb {R}^d\) to blow-up limits \(u_0,v_0\in C^{0,1}(\mathbb {R}^d)\). Then
-
(1)
taking \(C_{1}\) and \(C_{2}\) to be the constants from (1.2), we have
$$\begin{aligned} C_1v_0\le u_0\le C_2v_0\quad \text {on }\mathbb {R}^d; \end{aligned}$$ -
(2)
the functions \(u_0\) are \(v_0\) are harmonic in the open set \(\Omega _0:=\{u_0>0\}=\{v_0>0\}\).
-
(3)
there are constants \({\varepsilon }_0>0\) and \(C>0\) such that
$$\begin{aligned} {\varepsilon }_0 |B_r|\le |B_r(x_0)\cap \Omega _0|\le (1-{\varepsilon }_0)|B_r|, \end{aligned}$$(4.5)for every \(x_0\in \partial \Omega _0\), \(r>0\), and
$$\begin{aligned} |\{0<u_0<rt\}\cap B_r(x_0)|\leqq Ct|B_r|, \end{aligned}$$(4.6)for every \(x_0\in \partial \Omega _0, r>0,t>0\);
-
(4)
\(0\in \partial \Omega _0\) and there is a constant \(C_0 > 0\) such that
$$\begin{aligned} \sup _{B_r(x_0)} u_0 \ge C_0 r\quad \text {and}\quad \sup _{B_r(x_0)} v \ge C_0 r\quad \text {for every}\quad x_0\in {\overline{\Omega }}_0\quad \text {and every}\quad r>0; \end{aligned}$$ -
(5)
there is a constant \(\Lambda >0\) depending on \(C_1,C_2,C_Q\) and d such that, for every \(R>0\),
$$\begin{aligned} \left| \int _{B_{R}}\nabla u_0\cdot \nabla \varphi \,dx\right| +\left| \int _{B_{R}}\nabla v_0\cdot \nabla \varphi \,dx\right| \le \Lambda R^{d-1}\Vert \varphi \Vert _{L^\infty (B_{R})}; \end{aligned}$$ -
(6)
for every compactly supported smooth vector field \(\xi \in C^\infty _c(\mathbb {R}^d;\mathbb {R}^d)\), we have
$$\begin{aligned} \int _{\mathbb {R}^d} \Big (- \nabla u_{0} \cdot \big ((\nabla \xi ) + (D\xi )\big )\nabla v_{0}+\big (\nabla u_{0}\cdot \nabla v_{0}+ Q(x_0)\mathbbm {1}_{\Omega _0} \big ){{\,\textrm{div}\,}}\xi \Big )\, dx=0. \end{aligned}$$
Proof
For simplicity, we set
and we notice that
The first two claims are just a consequence of the locally uniform convergence of \(u_k\) and \(v_k\) to \(u_0\) and \(v_0\). By Proposition 3.1, Corollary 3.3 and Lemma 3.4, we already know that u and v fulfill the assumptions of Lemma 4.2. Therefore, (4) and (5) follow from Lemma 4.1, while (6) follows from Lemma 2.6 and Remark 2.7, and the strong \(H^1\) convergence of \(u_k\) and \(v_k\).
We next prove (3). By Proposition 3.5 and rescaling, we know that, for every \(k>0\),
so, by the strong convergence of \(\mathbbm {1}_{\Omega _k}\) to \(\mathbbm {1}_{\Omega _0}\) in \(L^1_{loc}(\mathbb {R}^d)\) (see Lemma 4.1), we get the density estimate (4.5) for \(\Omega _0\). Similarly, by rescaling (3.4) we obtain
which, passing to the limit as \(k\rightarrow \infty \), gives (4.6). \(\square \)
4.3 Second Blow-Up
Consider blow-up limits \(u_{0},v_{0}\) as in the previous subsection and let
be blow-up limits of \(u_0\) and \(v_0\) in zero. Then \(u_{00}\) and \(v_{00}\) are still blow-up limits of the state functions \(u_\Omega \) and \(v_\Omega \) at \(x_0\in \partial \Omega \cap D\) and Proposition 4.3 still applies. On the other hand, [29, Theorem 1.2] applies to the domain \(\Omega _0\) and the function \(u_0\), so the Boundary Harnack Principle (see [29, Definition 1.1]) holds on \(\Omega _0\); since \(u_0\) and \(v_0\) are harmonic in \(\Omega _0\), we get that the ratio \(u_0/v_0\) is Hölder continuous up to the boundary \(\partial \Omega _0\). This, in particular means that the second blow-ups \(u_{00}\) and \(v_{00}\) at any boundary point (and thus in zero) are proportional.
Lemma 4.4
Let \(u_0,v_0:\mathbb {R}^d\rightarrow \mathbb {R}\) be non-negative Lipschitz functions on \(\mathbb {R}^d\) with the same positivity set and let \(\Omega _0:=\{u_0>0\}=\{v_0>0\}\). Suppose that \(u_0\) and \(v_0\) satisfy the conditions (1)-(6) from Proposition 4.3 and let \(u_{00},v_{00}:\mathbb {R}^d\rightarrow \mathbb {R}\) be blow-ups of \(u_0,v_0\) at zero. Then
-
(1)
there is a constant \(\lambda \in (C_1,C_2)\) such that \(u_{00}=\lambda v_{00}\) on \(\mathbb {R}^d\);
-
(2)
the function \(u_{00}\) is harmonic in the open set \(\Omega _{00}:=\{u_{00}>0\}\);
-
(3)
there are constants \({\varepsilon }_0>0\) and \(C>0\) such that
$$\begin{aligned} {\varepsilon }_0 |B_r|\leqq |B_r(x_0)\cap \Omega _{00}| \leqq (1-{\varepsilon }_0)|B_r|, \end{aligned}$$for every \(x_0\in \partial \Omega _{00}\), \(r>0\), and
$$\begin{aligned} |\{0<u_{00}<rt\}\cap B_r(x_0)|\leqq Ct|B_r|, \end{aligned}$$for every \(x_0\in \partial \Omega _{00}, r>0,t>0\);
-
(4)
\(0\in \partial \Omega _{00}\) and there is a constant \(C_0 > 0\) such that
$$\begin{aligned} \sup _{B_r(x_0)}u_{00} \ge C_0 r\quad \text {for every}\quad x_0\in {\overline{\Omega }}_{00}\quad \text {and every}\quad r>0; \end{aligned}$$ -
(5)
there is a constant \(\Lambda >0\) such that, for every \(R>0\),
$$\begin{aligned} \left| \int _{B_{R}}\nabla u_{00}\cdot \nabla \varphi \,dx\right| \le \Lambda R^{d-1}\Vert \varphi \Vert _{L^\infty (B_{R})}; \end{aligned}$$ -
(6)
for every compactly supported smooth vector field \(\xi \in C^\infty _c(\mathbb {R}^d;\mathbb {R}^d)\), we have
$$\begin{aligned} \int _{\mathbb {R}^d} \bigg (- \nabla u_{00} \cdot \big ((\nabla \xi ) + (D\xi )\big )\nabla u_{00}+\Big (|\nabla u_{00}|^2+ {\lambda }Q(x_0)\mathbbm {1}_{\Omega _{00}} \Big ){{\,\textrm{div}\,}}\xi \bigg )\, dx=0. \end{aligned}$$(4.7)
Proof
Let \(R>0\) be fixed and let \(\phi :=u_0\). In order to show that the Boundary Harnack Principle holds on \(\Omega _0\), we check that \(\Omega _0\) and \(\phi \) satisfy the list of assumptions (a)–(g) from [29, Theorem 1.2] in the ball \(B_R\):
-
(a)
by definition of \(\Omega _0\), we have \(\phi >0\) in \(\Omega _0\) and \(\phi \equiv 0\) on \(B_R\setminus \Omega _0\);
-
(b)
by hypothesis \(\phi \) is Lipschitz continuous in \(\mathbb {R}^d\) and so, in \(B_R\);
-
(c)
since \(\phi \) is harmonic in \(\Omega _0\) and satisfies the condition (4) from Proposition Proposition 4.3, we can apply [38, Lemma 6.8]; thus, there is a constant \(\kappa >0\) such that
$$\begin{aligned} \phi \ge \kappa \,{{\,\textrm{dist}\,}}_{B_R\setminus \Omega _0}\quad \text {in }B_{{R}/{2}}; \end{aligned}$$ -
(d)
since \(\phi \geqq 0\) and \(\Delta \phi =0\) in \(\Omega _0\), we have that \(\Delta \phi \ge 0\) in \(\mathbb {R}^d\);
-
(e)
for every \(x_0\in \partial \Omega \cap B_{R}\), we have
$$\begin{aligned} |B_r(x_0)\setminus \Omega _0|\ge {\varepsilon }_0 |B_r(x_0)|\qquad \text {for every }r\in (0,R-|x_0|); \end{aligned}$$ -
(f)
for every \(x_0\in \partial \Omega _0\cap B_{R}\) and every \(r\in (0,R-|x_0|)\), we have
$$\begin{aligned} \big |\{0<\phi <rt\}\cap B_{r}(x_0)\big |\le C t|B_r|\qquad \text {for every }t>0;. \end{aligned}$$ -
(g)
by (4) of Proposition 4.3, for every \(x_0\in \partial \Omega _0\cap B_{R}\) and every \(r\in (0,R-|x_0|)\), we have
$$\begin{aligned} \sup _{B_r(x_0)}\phi \ge C_0 r. \end{aligned}$$
Therefore, all the assumptions of [29, Theorem 1.2] are fulfilled and so, the ratio
can be extended to a Hölder continuous function on \({\overline{\Omega }}_0\cap B_{{R}/{2}}\). Thus, the blow-ups \(u_{00}\) and \(v_{00}\) are proportional and all the other claims follow as in Proposition 4.3. \(\square \)
4.4 Third Blow-Up
Take the state functions \(u_\Omega \) and \(v_\Omega \) on an optimal domain \(\Omega \). Let
be the second blow-up limits of \(u_\Omega \) and \(v_\Omega \) at a free boundary point \(x_0\in \partial \Omega \cap D\). Then, \(u_{00}\) and \(v_{00}\) are proportional and satisfy the conditions listed in Lemma 4.4. We will show that if we perform a further blow-up in zero, then we obtain functions \(u_{000},v_{000}:\mathbb {R}^d\rightarrow \mathbb {R}\) that still satisfy the conditions from Lemma 4.4 but are also 1-homogeneous. Before we give the precise statement, we notice that the stationarity condition (4.7) implies the monotonicity of the associated Weiss’ boundary adjusted energy from [39].
Lemma 4.5
(Monotonicity formula) Let \(B_R\subset \mathbb {R}^d\), \(u \in H^1(B_R)\) be a continuous non-negative function, and let \(\Omega :=\{u>0\}\). Suppose that for every smooth compactly supported vector field \(\xi \in C^\infty _c(\mathbb {R}^d;\mathbb {R}^d)\), we have:
Then, for every \(x_0 \in \partial \Omega , r>0\) the map
is non-decreasing in \((0,+\infty )\) and
In particular, if \(r\mapsto W_\Lambda (u_{x_0,r})\) is constant, then u is 1-homogeneous.
Proof
See for instance [38, Proposition 9.9]. \(\square \)
Lemma 4.6
Let \(u_{00},v_{00}:\mathbb {R}^d\rightarrow \mathbb {R}\) be non-negative Lipschitz functions on \(\mathbb {R}^d\) with the same positivity set and let \(\Omega _{00}:=\{u_{00}>0\}=\{v_{00}>0\}\). Suppose that \(u_{00}\) and \(v_{00}\) satisfy the conditions (1)-(6) from Lemma 4.4 and let \(u_{000},v_{000}:\mathbb {R}^d\rightarrow \mathbb {R}\) be blow-ups of \(u_{00},v_{00}\) at zero. Then:
-
(1)
\(u_{000}=\lambda v_{000}\) on \(\mathbb {R}^d\);
-
(2)
for every compactly supported smooth vector field \(\xi \in C^\infty _c(\mathbb {R}^d;\mathbb {R}^d)\), we have
$$\begin{aligned} \int _{\mathbb {R}^d} \bigg (- \nabla u_{000} \cdot \big ((\nabla \xi ) + (D\xi )\big )\nabla u_{000}+\Big (|\nabla u_{000}|^2+ {\lambda }Q(x_0)\mathbbm {1}_{\Omega _{000}} \Big ){{\,\textrm{div}\,}}\xi \bigg )\, dx=0;\nonumber \\ \end{aligned}$$(4.8) -
(3)
\(u_{000}\) is 1-homogeneous in \(\mathbb {R}^d\).
Proof
Set \(\Lambda :={\lambda }Q(x_0)\) and for simplicity, let \(u:=u_{00}\). By Lemma 4.5, the function
is non-decreasing in r and so, it admits a limit \(\Theta \) as \(r\rightarrow 0\); moreover, by the Lipschitz continuity of u, \(\Theta \) is finite. Let \(r_k\rightarrow 0\) be such that \(u_{r_k}\rightarrow u_{000}\). Then, by Lemma 4.1, \(u_{r_k}\) converges to \(u_{000}\) strongly in \(H^1_{loc}\) and the level sets \(\{u_{r_k}>0\}\) converge in \(L^1_{loc}\) to \(\Omega _{000}\). Thus, for any \(s>0\)
Moreover, using again the strong convergence of \(u_{r_k}\) and their level sets, we get that \(u_{000}\) satisfies (4.8). Thus, using again Lemma 4.5 and the fact that \(s\mapsto W_\Lambda \big ((u_{000})_s\big )\) is constantly equal to \(\Theta \), we get that \(u_{000}\) is homogeneous. \(\square \)
As an immediate consequence, we obtain the following proposition.
Proposition 4.7
(Existence of stationary 1-homogeneous blow-ups) Let \(D\subset \mathbb {R}^d\) be a bounded open set and \(\Omega \) be a solution to (1.3) with f, g, Q as in Theorem 1.2. Let \(u:=u_\Omega \) and \(v:=v_\Omega \) be the state functions on \(\Omega \) defined in (1.1) and (1.6) and let \(x_0 \in \partial \Omega \cap D\). Then, there is a sequence \(r_k\rightarrow 0\) such that the corresponding blow-up limits
satisfy the following conditions:
-
(i)
\(u_0\) and \(v_0\) are 1-homogeneous in \(\mathbb {R}^d\) and \(u_{0}=\lambda v_{0}\) for some constant \(\lambda \in (C_1,C_2)\);
-
(ii)
for every compactly supported smooth vector field \(\xi \in C^\infty _c(\mathbb {R}^d;\mathbb {R}^d)\), we have
$$\begin{aligned} \int _{\mathbb {R}^d}\bigg (-\nabla u_0\cdot \big ((\nabla \xi )+(D\xi )\big )\nabla u_0+\Big (|\nabla u_0|^2+\lambda Q(x_0)\mathbbm {1}_{\Omega _0}\Big ){{\,\textrm{div}\,}}\xi \bigg )\,dx=0. \end{aligned}$$
Remark 4.8
The blow-up sequence arising from Proposition 4.7 is obtained by a diagonal argument between the three different blow-ups defined respectively in Sects. 4.2, 4.3 and 4.4. Therefore, by combining all the previous result, we can see that \(u_0\) and \(v_0\) fulfill the conditions (a), (b), (c), d and (e) in Definition 7.3.
5 Regular and Singular Parts of the Free Boundary
Let \(D\subset \mathbb {R}^d\) be a bounded open set and let \(\Omega \) be a solution to the problem (1.3). As in Sect. 4, we denote by u, v the associated state variables.
Definition 5.1
(Regular and singular points) We will say that a boundary point \(x_0\in \partial \Omega \cap D\) is regular if there is a blow-up limit \((u_{0},v_0)\) of (u, v) at \(x_0\) such that
for some unit vector \(\nu \in \mathbb {R}^d\) and some \(\alpha >0\) and \(\beta >0\) such that \(\alpha \beta = Q(x_0)\). If such a blow-up limit does not exist, then we will say that \(x_0\) is singular.
We will denote by \(\text { Reg}(\partial \Omega )\) the set of all regular points on \(\partial \Omega \cap D\) and by \(\text { Sing}(\partial \Omega )\) the set of all singular points on \(\partial \Omega \cap D\). Clearly, we have that
In Sect. 6 we will show that the regular part \(\text { Reg}(\partial \Omega )\) is in fact, locally, a smooth manifold (and in particular, a relatively open subset of \(\partial \Omega \cap D\)), while in Sect. 8 we will give an estimate on the dimension of the singular set \(\text { Sing}(\partial \Omega )\).
5.1 Regular and Singular Parts in Dimension Two
One can easily show that in dimension \(d=2\) the free boundary \(\partial \Omega \cap D\) is composed only of regular points; in particular, in dimension two the proof of Theorem 1.2 is concluded already in Sect. 6, while the results from Sect. 8 are needed only when \(d\ge 3\).
Lemma 5.2
Let D be a bounded open set in \(\mathbb {R}^2\) and let \(\Omega \) be a solution to (1.3). Then, every point \(x_0 \in \partial \Omega \cap D\) is a regular point in the sense of Definition 5.1.
Proof
Let \(r_k\rightarrow 0\) be a sequence such that the blow-up limits
are, as in Proposition 4.7, proportional (\(u_0=\lambda v_0\)), non-negative and 1-homogeneous functions on \(\mathbb {R}^d\), which are harmonic on the positivity set \(\Omega _0:=\{u_0>0\}=\{v_0>0\}\). Now, reasoning as in [38, Proposition 9.13], we write \(u_0\) and \(v_0\) in polar coordinates as
where (since \(u_0\) and \(v_0\) are harmonic) \(\phi \) and \(\psi \) are solutions to
Since the only solutions of this equations (up to a rotation) are multiples of \(\sin \theta \), and since \(\mathcal {H}^{1}(\Omega _0\cap \partial B_1)<2\pi \) (this follows from the density estimate in Proposition 4.3), we get that
for some unit vector \(\nu \in \mathbb {R}^d\) and some \(\alpha >0\) and \(\beta >0\). Moreover, by Proposition 4.7, the function \(u_0\) is critical point for the one-phase problem (that is, (4.8) holds). Thus,
where \(\lambda =\alpha /\beta \). Since \(|\nabla u_0|^2=\alpha ^2\), we get that \(\alpha \beta =Q(x_0)\), which concludes the proof. \(\square \)
5.2 A Geometric Condition for the Regularity in Every Dimension
By an argument similar to the one in Lemma 5.2, we have that the free boundary points admitting one-sided tangent ball are regular points. This result holds in every dimension and will be useful in Sect. 6.
Lemma 5.3
Let D be a bounded open set in \(\mathbb {R}^d\), \(d\ge 2\), and let \(\Omega \) be a solution to (1.3). Suppose that there is a one-sided tangent ball at the boundary point \(x_0 \in \partial \Omega \cap D\) in the sense that:
Then, \(x_0\) is a regular point in the sense of Definition 5.1.
Proof
As in Lemma 5.2, we consider a sequence \(r_k\rightarrow 0\) for which the blow-up limits
are 1-homogeneous, non-negative, harmonic on their positivity set \(\Omega _0:=\{u_0>0\}\) and are such that \(u_0=\lambda v_0\). The one-sided ball condition implies that there is a unit vector \(\nu \in \mathbb {R}^d\) such that
Since \(\Omega _0\) satisfies an exterior density estimate (by Proposition 4.3, claim Item (3)), the only possibility is that \(\Omega _0=\{x\in \mathbb {R}^d:\,x\cdot \nu >0\}\) and
for some \(\alpha >0\) and \(\beta >0\) with \(\alpha /\beta =\lambda \). As in Lemma 5.2, since \(u_0\) satisfies (4.8), we get that
which gives that \(\alpha \beta =Q(x_0)\). \(\square \)
6 Regularity of \(\text { Reg}(\partial \Omega )\)
In this section we prove that the regular part \(\text { Reg}(\partial \Omega )\) of the boundary of an optimal set \(\Omega \) is locally the graph of a smooth function.
6.1 Viscosity Formulation
In this subsection we prove that on the free boundary \(\partial \Omega \cap D\) of an optimal set \(\Omega \), solution to (1.3), we have the following optimality condition
in viscosity sense, as in [28, Section 2], in terms of the blow-up limit of the state functions \(u_\Omega \) and \(v_\Omega \) at free boundary points (see Remark 6.2).
Definition 6.1
(Viscosity solutions) Let D be a bounded open set in \(\mathbb {R}^d\) and let \(f,g\in L^\infty (\mathbb {R}^d)\). Let \(u,v :D\rightarrow \mathbb {R}\) be two non-negative continuous functions with the same support
on which they satisfy the PDE
We say that the boundary condition
holds in viscosity sense if at any point \(x_0\in \partial \Omega \cap D\), at which \(\Omega \) admits a one-sided tangent ball in the sense of (5.1), there exist:
-
(a)
a decreasing sequence \(r_k\rightarrow 0\);
-
(b)
two positive constants \(\alpha ,\beta >0\) such that \(\alpha \beta =Q(x_0)\);
-
(c)
a unit vector \(\nu \in \mathbb {R}^d\);
such that the rescalings
converge uniformly in every ball \(B_R\subset \mathbb {R}^d\) respectively to the blow-up limits
Remark 6.2
In [28] the authors addressed the \({\varepsilon }\)-regularity theory for viscosity solutions of (1.8). In particular, in [28, Lemma 2.9] they proved that if the free boundary condition is satisfied in the sense of Definition 6.1, then for any smooth function \(\varphi \in C^\infty (D)\) the following holds:
-
(i)
If \(\varphi _+\) touches \(\sqrt{uv}\) from below at a point \(x_0\in D\cap \partial \Omega \), then \(|\nabla \varphi (x_0)|\le \sqrt{Q(x_0)}\).
-
(ii)
If \(\varphi _+\) touches \(\sqrt{uv}\) from above at a point \(x_0\in D\cap \partial \Omega \), then \(|\nabla \varphi (x_0)|\ge \sqrt{Q(x_0)}\).
-
(iii)
If a and b are constants such that
$$\begin{aligned} a>0,\quad b>0\qquad \text {and}\qquad a b=Q(x_0), \end{aligned}$$and if \(\varphi _+\) touches \(w_{ab}:=\frac{1}{2}(au+bv)\) from above at \(x_0\in D\cap \partial \Omega \), then \(|\nabla \varphi (x_0)|\ge \sqrt{Q(x_0)}\).
Proposition 6.3
Let \(D\subset \mathbb {R}^d\) be a bounded open set and let \(f,g,Q:D\rightarrow \mathbb {R}\) be as in Theorem 1.2. Let \(\Omega \) be a solution to (1.3). Then the state variables \(u:=u_\Omega \) and \(v:=v_\Omega \) satisfy
in the sense of Definition 6.1.
Proof
It follows as in the proof of Lemma 5.3. \(\square \)
We can now prove that \(\text { Reg}(\partial \Omega )\) is \(C^{1,\alpha }\)-regular for some \(\alpha \in (0,1)\) by exploiting the \({\varepsilon }\)-regularity theory developed in [28].
Theorem 6.4
(Regularity of \(\text { Reg}(\partial \Omega )\)) Let D be a bounded open set in \(\mathbb {R}^d\), where \(d\ge 2\). Let
be given non-negative functions. Suppose that the following conditions hold:
-
(a)
\(f,g\in L^\infty (D)\);
-
(b)
there are constants \(C_1,C_2 > 0\) such that
$$\begin{aligned} 0\le C_1 g\le f\le C_2 g\qquad \text{ in } D. \end{aligned}$$ -
(c)
\(Q\in C^{0,\alpha _Q}(D)\), for some \(\alpha _Q>0\), and there are a positive constants \(c_Q,C_Q\) such that
$$\begin{aligned} 0<c_Q\le Q(x)\leqq C_Q\qquad \text{ for } \text{ every } x\in D. \end{aligned}$$
Let \(\Omega \) be a solution to (1.3). Then, the regular part \(\text { Reg}(\partial \Omega )\), defined in Sect. 5, is locally the graph of a \(C^{1,\alpha }\) function, for some \(\alpha >0\).
Proof
The proof follows by the epsilon-regularity theory developed in [28]. Indeed, if \(x_0 \in \text { Reg}(\partial \Omega )\) and \(u_{x_0,\rho _k}, v_{x_0,\rho _k}\) are the blow-up sequences from the definition of \(\text { Reg}(\partial \Omega )\). Then, there are \(\alpha ,\beta >0, \nu \in \mathbb {R}^d\) such that \(\alpha \beta =Q(x_0), |\nu |=1\) and
Moreover, the Lipschitz continuity and the non-degeneracy imply that for every \({\varepsilon }>0\) there exists \(k_0>0\) such that for every \(k\geqq k_0\) we have
that is, \(u_{x_0,\rho _k}, v_{x_0,\rho _k}\) are \({\varepsilon }\)-flat in the direction \(\nu \) (see [28, Definition 1.2]). By rescaling the associated state equations, we have
where
On the other hand, since both \(u_{x_0,\rho _k}\) and \(v_{x_0,\rho _k}\) still satisfy (6.3) in viscosity sense, by applying [28, Theorem 3.1], \(\partial \{u_{x_0,\rho _k}>0\}\) is \(C^{1,\alpha }\) in \(B_{1/2}\). Finally, the result follows by rescaling back to the original problem. \(\square \)
6.2 Higher Regularity
We can pass from \(C^{1,\alpha }\) to \(C^\infty \)-regularity of \(\text { Reg}(\partial \Omega )\) by exploiting the higher order Boundary Harnack Principle for solutions to (1.8).
Proposition 6.5
(Higher regularity of \(\text { Reg}(\partial \Omega )\)) Let D be a bounded open set in \(\mathbb {R}^d\) and let f, g, Q be as in Theorem 6.4. If \(f,g,Q\in C^{k,\alpha }(D)\) for some \(k\ge 1\) and \(\alpha >0\), then the regular part \(\text { Reg}(\partial \Omega )\) of the free boundary \(\partial \Omega \cap D\) of any solution \(\Omega \) to (1.3) is locally the graph of a \(C^{1+k,\alpha }\) function for some \(\alpha >0\). In particular, if f, g, Q are \(C^\infty \), then \(\text { Reg}(\partial \Omega )\) is locally the graph of a \(C^\infty \) function.
Proof
We use a bootstrap argument as in [31, Section 5.4]. Suppose that \(\text { Reg}(\partial \Omega )\) is locally the graph of a \(C^{k,\alpha }\)-regular function, for some \(k\geqq 1\) (when \(k=1\) the claim follows from Theorem 6.4). Let \(x_0 \in \partial \Omega \) and \(r>0\) be such that \(\partial \Omega \cap B_r(x_0)= \text { Reg}(\partial \Omega )\). Since u and v satisfy (6.1) in \(\Omega \) and since v is non-degenerate, we can apply the higher order Boundary Harnack Principle (for PDEs with right-hand side) from [27, Theorem 1.3] and [36, Theorem 1.3], obtaining a non-negative \(C^{k,\alpha }\)-regular function \(w:B_r(x_0)\cap \overline{\Omega }\rightarrow \mathbb {R}\) satisfying \(u =w v\) in \(B_r(x_0)\cap \overline{\Omega }\). Then, we have:
Thus, u is a solution to the problem
which by [24, Theorem 2] implies that \(\text { Reg}(\partial \Omega )\) is locally the graph of a \(C^{k+1,\alpha }\) function. \(\square \)
7 Stable Homogeneous Solutions of the One-Phase Bernoulli Problem
In this section we study the singular set of the stable global solutions of the one-phase Bernoulli problem (see Definition 7.3 for the definition of global stable solution). The main results are Theorems 7.8 and 7.9, which we will use in Theorem 8.1 in order to estimate the dimension of the singular set of the free boundary of the optimal sets for (1.3). The present section can be read separately from the rest of the paper; we will use only the Taylor expansions from Sect. 2 and the general results from Sect. 4.
7.1 Solutions of PDEs in Unbounded Domains
In \(\mathbb {R}^d\), \(d\ge 3\), we define
It is well known that \(\dot{H}^1(\mathbb {R}^d)\) is a Hilbert space equipped with the norm \(\Vert u\Vert _{\dot{H}^1(\mathbb {R}^d)}:=\Vert \nabla u\Vert _{L^2(\mathbb {R}^d;\mathbb {R}^d)}\) and that \(C^\infty _c(\mathbb {R}^d)\) is dense in \(\dot{H}^1(\mathbb {R}^d)\). Moreover, given an open (bounded or unbounded) set \(\Omega \subset \mathbb {R}^d\), we define the space \(\dot{H}^1_0(\Omega )\) as the closure of \(C^\infty _c(\Omega )\) with respect to \(\Vert \cdot \Vert _{\dot{H}^1(\mathbb {R}^d)}\).
Let \(\Omega \) be an open (bounded or unbounded) subset of \(\mathbb {R}^d\) and let \(F\in L^2(\mathbb {R}^d;\mathbb {R}^d)\) be a given vector field. We say that w is a weak solution of the PDE
if \(w\in \dot{H}^1_0(\Omega )\) and
It is standard to check that w is a solution to (7.1) if and only if w minimizes the functional
among all functions \(\varphi \in \dot{H}^1_0(\Omega )\). Since it is immediate to check that a minimizer of (7.2) in \(\dot{H}^1_0(\Omega )\) exists and is unique, we get that also the solution to (7.1) exists and is unique. Finally, we notice that if \(w\in \dot{H}^1_0(\Omega )\) is the solution to (7.1), then
which gives
Remark 7.1
(Exterior density estimate and the space \(\dot{H}^1_0\)) We say that an open set \(\Omega \subset \mathbb {R}^d\) satisfies a uniform exterior density estimate with a constant \(c>0\) if
It is known (see for example [17]) that if an open set \(\Omega \subset \mathbb {R}^d\) satisfies (7.4) then the space \(\dot{H}^1_0(\Omega )\) can be characterized as:
Lemma 7.2
(Convergence of solutions) Let \(\Omega _n\) be a sequence of open sets in \(\mathbb {R}^d\), \(d\ge 3\) such that:
-
(a)
there is a constant \(c>0\) such that, for every \(n\ge 1\), \(\Omega _n\) satisfies the exterior density estimate (7.4);
-
(b)
there is an open set \(\Omega _\infty \subset \mathbb {R}^d\) satisfying the exterior density estimate (7.4) with the constant \(c>0\) and such that the sequence of characteristic functions \(\mathbbm {1}_{\Omega _n}\) converges pointwise almost-everywhere in \(\mathbb {R}^d\) to \(\mathbbm {1}_{\Omega _\infty }\).
Let \(F_n\in L^2(\mathbb {R}^d;\mathbb {R}^d)\) be a sequence of vector fields converging strongly in \(L^2(\mathbb {R}^d;\mathbb {R}^d)\) to the vector field \(F_\infty \in L^2(\mathbb {R}^d;\mathbb {R}^d)\). For every \(n\ge 1\), let \(w_n\) be the solution to the PDE
Then, \(w_n\) converges strongly in \(\dot{H}^1(\mathbb {R}^d)\) to the solution \(w_\infty \) of
Proof
First of all, we notice that the sequence \(w_n\) is bounded in \(\dot{H}^1(\mathbb {R}^d)\) (by (7.3)). Thus, we can extract a subsequence, that we still denote by \(w_{n}\), which converges weakly in \(\dot{H}^1(\mathbb {R}^d)\) and pointwise almost-everywhere on \(\mathbb {R}^d\) to a function \(w\in \dot{H}^1(\mathbb {R}^d)\). We will show that \(w=w_\infty \).
First, notice that for almost every \(x\in \mathbb {R}^d\setminus \Omega _\infty \) we have that
But then \(w_n(x)=0\) (by (7.5)) and so \(w(x)=0\). Thus, using again (7.5), we get \(w\in \dot{H}^1_0(\Omega _\infty )\).
Now, let \(\varphi \in C^\infty _c(\Omega _\infty )\). We will show that for large enough n, \(\varphi \in C^\infty _c(\Omega _n)\). Indeed, let \(\delta >0\) be a constant such that \(B_\delta (x)\subset \Omega _\infty \) for every x in the support of \(\varphi \). Suppose by contradiction that there is a sequence \(x_n\in \overline{\{\varphi \ne 0\}}\) such that \(x_n\notin \Omega _n\). Then, by the density estimate for \(\Omega _n\), \(|B_\delta (x_n)\cap \Omega _n|\le (1-c)|B_{\delta }|\). Now, up to a subsequence, \(x_n\rightarrow x_\infty \in \overline{\{\varphi \ne 0\}}\). But then,
which is a contradiction. Thus, for large n, \(\overline{\{\varphi \ne 0\}}\subset \Omega _n\). Now, using the equation for \(w_n\),
and passing to the limit, we get that
that is \(w=w_\infty \).
Finally, in order to prove that the convergence is strong, we use the equations for \(w_n\) and \(w_\infty \) and the strong convergence of \(F_n\) to \(F_\infty \):
which concludes the proof. \(\square \)
7.2 Global Stable Solutions of the One-Phase Problem
We define the functionals
as follows. Given:
-
a Lipschitz function \(u:\mathbb {R}^d\rightarrow \mathbb {R}\),
-
a smooth compactly supported vector field \(\xi \in C^\infty _c(\mathbb {R}^d;\mathbb {R}^d)\),
we set:
where \(\Omega _u:=\{u>0\}\), and where \(\delta A,\,\delta ^2A\) are defined as in (2.6) and where \(\delta u\in \dot{H}^1_0(\Omega _u)\) is the weak solution to the PDE
Definition 7.3
(Global stable solutions of the one-phase problem) We say that a function \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) is a global stable solution of the one-phase problem if, for every compactly supported smooth vector field \(\xi \in C^\infty _c(\mathbb {R}^d;\mathbb {R}^d)\), we have:
and if the following conditions hold:
-
(a)
u is globally Lipschitz continuous and non-negative on \(\mathbb {R}^d\);
-
(b)
u is harmonic in the open set \(\Omega _u:=\{u>0\}\);
-
(c)
there is a constant \(c>0\) such that
$$\begin{aligned} |B_r(x_0)\cap \Omega _u| \leqq (1-c)|B_r|, \end{aligned}$$for every \(x_0\in \mathbb {R}^d\setminus \Omega _u\) and every \(r>0\);
-
(d)
\(0\in \partial \Omega _u\) and there is a constant \(\eta > 0\) such that
$$\begin{aligned} \sup _{B_r(x_0)} u \ge \eta r\quad \text {for every}\quad x_0\in {\overline{\Omega }}_{u}\quad \text {and every}\quad r>0; \end{aligned}$$ -
(e)
there is a constant \(C>0\) such that, for every \(R>0\),
$$\begin{aligned} \left| \int _{B_{R}}\nabla u\cdot \nabla \varphi \,dx\right| \le C R^{d-1}\Vert \varphi \Vert _{L^\infty (B_{R})}\quad \text {for every}\quad \varphi \in C^\infty _c(B_R). \end{aligned}$$
Remark 7.4
The functionals \(\delta \mathcal {G}\) and \(\delta ^2\mathcal {G}\) correspond to the first and the second variation along vector fields of the one-phase Alt–Caffarelli functionalFootnote 1
so one may expect that the natural definition of a global stable solution is a function \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) that satisfies (7.8). Unfortunately, the condition (7.8) alone seems to be quite weak. For instance,
satisfy (7.8), but they are not even harmonic in \(\{u>0\}\), while the function
is harmonic in \(\{u>0\}\), but the free boundary \(\partial \{u>0\}\) has a cross-like singularity in zero; more generally, the solutions of optimal partition problems satisfy (7.8) and the structure of their nodal sets can be very different from the one of the one-phase free boundaries.
We add the conditions (a), (b), (c), d, (e) in order to have a regularity theory for one-phase stable solutions, which is similar to the one available for minimizers of the Alt–Caffarelli functional. The Lipschitz continuity (a) and the non-degeneracy d guarantee the existence of non-trivial blow-up limits obtained by 1-homogeneous rescalings of u. The condition (e) is needed for the strong convergence of the blow-up sequences (see Lemma 4.1), which together with the exterior density estimate (c) allows to transfer the stability condition (7.8) to the blow-up limits of u (thanks to Lemma 7.2). We also notice that the conditions (e) and (b) imply the Lipschitz continuity of u, so we could actually avoid adding (a) to the list.
Finally, we highlight that the conditions (a)–(b)–(c)-d–(e) are satisfied by the blow-ups of numerous one-phase problems, for instance by the state functions on the domains minimizing (1.3) (see Sect. 4), and of course, by the global minimizers of the classical one-phase Bernoulli problem.
Remark 7.5
(Blow-ups of global stable solutions) We notice that if \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) satisfies (a), (b), (c), d and (e), then any blow-up \(u_0:\mathbb {R}^d\rightarrow \mathbb {R}\) of u at \(x_0\in \partial \Omega _u\),
still satisfies (a), (b), (c), d and (e). In particular, by Lemma 4.1, this means that the convergence \(u_{x_0,r_n}\rightarrow u_0\) is strong in \(H^1_{loc}\) and thus, by Lemmas 7.2 and 4.5, if u is a global stable solution, then \(u_0\) is a 1-homogeneous global stable solution.
Remark 7.6
(Decomposition of the free boundary) Let \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) be a global stable solution in the sense of Definition 7.3 and let \(\Omega _u:=\{u>0\}\). We decompose the free boundary \(\partial \Omega _u\) as
where the regular part \(\text { Reg}(\partial \Omega _u)\) consists of all points \(x_0\in \partial \Omega _u\) at which there is a blow-up limit \(u_0\) which is a half-space solution, that is,
while the singular part is given by \(\text { Sing}(\partial \Omega _u)=\partial \Omega _u\setminus \text { Reg}(\partial \Omega _u)\). As in Sect. 6, it is immediate to check that the global stable solutions satisfy the optimality condition
in viscosity sense, so the \({\varepsilon }\)-regularity theorem of [18] holds and we have that the regular part is a relatively open subset of \(\partial \Omega _u\) and a \(C^\infty \) manifold. We notice that this decomposition is precisely the one from Sect. 5 with \(\alpha =\beta =Q=1\), \(u=v\), and \(f=g=0\).
Definition 7.7
(Critical dimension for stable solutions) We define \(d^*\) to be the smallest dimension admitting a 1-homogeneous global stable solution (in the sense of Definition 7.3) \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) with \(0\in \text { Sing}(\partial \Omega _u)\).
Theorem 7.8
(Dimension reduction for stable solutions) Suppose that \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) is a 1-homogeneous global stable solution of the one-phase problem (in the sense of Definition 7.3).
-
(i)
If \(d< d^*\), then u is a half-plane solution.
-
(ii)
If \(d=d^*\), then \(\text { Sing}(\partial \Omega _u)=\{0\}\) or u is a half-plane solution.
-
(iii)
If \(d>d^*\), then the Hausdorff dimension of \(\text { Sing}(\partial \Omega _u)\) is at most \(d-d^*\), that is,
$$\begin{aligned} \mathcal {H}^{d-d^*+{\varepsilon }}\big (\text { Sing}(\partial \Omega _u)\big )=0\quad \text {for every}\quad {\varepsilon }>0, \end{aligned}$$
where \(d^*\) is the critical dimension from Definition 7.7.
Proof
The proof follows from a classical argument that can be found for instance in [38] (in particular, [38, Proposition 10.13]). \(\square \)
Theorem 7.9
(Bounds on the critical dimension for stable solutions) \(5\le d^*\le 7\), where \(d^*\) is the critical dimension from Definition 7.7.
Proof
The claim follows from Propositions 7.11 and 7.12 below. \(\square \)
7.3 Global Minimizers and Global Stable Solutions
Given an open set \(D\subset \mathbb {R}^d\) and a function \(u\in H^1(D)\), we define:
Definition 7.10
(Global minimizers) We say that a function \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) is a global minimizer of the Alt–Caffarelli functional, if:
-
u is non-negative and \(u\in H^1_{loc}(\mathbb {R}^d)\);
-
\(\mathcal G(u,B_R)\le \mathcal G(v,B_R)\), for every \(B_R\subset \mathbb {R}^d\) and every \(v:\mathbb {R}^d\rightarrow \mathbb {R}\) such that \(v-u\in H^1_0(B_R)\).
Proposition 7.11
Suppose that \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) is a global minimizer in the sense of Definition 7.10. Then, u is a global stable solution in the sense of Definition 7.3. In particular, \(d^*\le 7\), where \(d^*\) is the critical dimension from Definition 7.7.
Proof
It is well-known that the global minimizers satisfy the conditions (a)-(b)-(c)-d-(e) of Definition 7.3 (see for instance [38]). Moreover, by [38, Lemma 9.5 and Lemma 9.6], the global minimizers are critical points, that is,
Thus, it only remains to prove the positivity of the second variation:
Let \(\xi \in C^\infty _c(\mathbb {R}^d;\mathbb {R}^d), \Phi _t\) be the associated defined by (2.5) and let \(\Omega _t:= \Phi _t(\Omega )\), for \(t\in \mathbb {R}\). In \(B_R\), we consider the solution \(u_t\) to the problem
By the optimality of \(u_t\), we have that
so,
By Lemma 2.8, we have that
where \(\Omega _u=\{u>0\}\) and \(w_R\) is the solution to the PDE
Thus, for every \(R>0\),
Since by Lemma 7.2 the sequence \(w_R\rightarrow \delta u\) strongly in \(\dot{H}^1(\mathbb {R}^d)\) as \(R\rightarrow \infty \), we get that
which is precisely the inequality \(\delta ^2\mathcal {G}(u)[\xi ]\ge 0\). Finally, the bound \(d^*\le 7\) follows by the example of a singular 1-homogeneous global minimizer constructed by De Silva and Jerison in [19]. \(\square \)
7.4 Global Stable Solutions and the Stability Inequality of Caffarelli–Jerison–Kenig
Let \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) be a 1-homogeneous global stable solution of the one-phase problem with an isolated singularity in zero, that is,
In particular, we have that the regular part
is a smooth \(C^\infty \) manifold and the function u is \(C^\infty \) in \({\overline{\Omega }}_u\setminus \{0\}\), up to the boundary \(\partial \Omega _u\setminus \{0\}\). Thus, u is a classical solution to the PDE
Together with the homogeneity of u this implies (see for instance [22]) that
where H is the mean curvature of \(\partial \Omega _u\) oriented towards the complement of \(\Omega _u\).
We will say that \(\Omega _u\) supports the stability inequality of Caffarelli-Jerison-Kenig if
In [14] and [22] it was shown that if:
-
\(u:\mathbb {R}^d\rightarrow \mathbb {R}\) is a 1-homogeneous non-negative Lipschitz function;
-
\(\partial \Omega _u\setminus \{0\}\) is \(C^\infty \) smooth;
-
u is a solution of the one-phase Bernoulli problem (7.9);
-
\(\Omega _u\) supports the stability inequality (7.10);
then u is a half space solution, that is,
Thus, in order to show that in dimension 3 and 4 there are no global stable solutions (in the sense of Definition 7.3) with singularities, it is sufficient to prove the following proposition.
Proposition 7.12
(The global stable cones satisfy the stability inequality) Let \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) be a 1-homogeneous global stable solution (in the sense of Definition 7.3) with \(\text { Sing}(\partial \Omega _u)=\{0\}\). Then, \(\Omega _u\) supports the stability inequality (7.10). In particular, \(d^*\ge 5\), where \(d^*\) is the critical dimension from Definition 7.7.
Remark 7.13
Following the proof of Proposition 7.12, it is immediate to check that also the converse is true. Precisely, if \(u:\mathbb {R}^d\rightarrow \mathbb {R}\) is a 1-homogeneous function, if \(\partial \Omega _u\setminus \{0\}\) is smooth and if u is a solution to (7.9) such that \(\Omega _u\) supports the stability inequality (7.10), then u is a global stable solution in the sense of Definition 7.3.
Proof
For any bounded open set \(D\subset \mathbb {R}^d\) and any function \(u\in H^1(D)\), we define:
Moreover, for any \(R>1\), we call the annulus
We fix a smooth vector field \(\xi \in C^\infty _c(A_R,\mathbb {R}^d)\) and we define the open set
where \(\Phi _t\) is the flow of \(\xi \) defined by (2.5). Let \(u_t:A_R\rightarrow \mathbb {R}\) be the solution of the PDE
Step 1. We will show that
where \(\delta ^2\mathcal {G}\) is defined in (7.6). Following Lemma 2.8 we have that
where \(\delta A\) and \(\delta ^2A\) are defined in (2.6) and \(w_R\) is the solution of the PDE
Now, let \(\delta u\) be the solution to (7.7). By the variational characterization of (7.7) in \(\dot{H}^1_0(\Omega _u)\), the fact that \( w_R\in \dot{H}^1_0(\Omega _u)\), and an integration by parts, we have that
which gives (7.11).
Step 2. We set \(u'\) to be the solution of the PDE
where \(\nu \) is the outer normal to \(\partial \Omega _u\). We will first show that
Indeed, since \(\xi \) is supported in \(A_R\) and since \(\nabla u=-\nu \) on \(\partial \Omega _u\cap A_R\), we have:
In order to show that \(u'\) is harmonic in \(\Omega _u\), we compute (using the repeated index summation convention)
and then we use the equation (7.12) for \(w_R\).
Step 3. We next compute the second derivative of \(\mathcal G(u_t,A_R)\) in terms of \(u'\). For the sake of simplicity, through the rest of the proof we use the notations introduced in Sect. 1.5.
On the other hand
Notice that
and
which leads to
Similarly, we can compute
and so
Finally, by collecting the previous computations, we obtain the identity
![](http://media.springernature.com/lw542/springer-static/image/art%3A10.1007%2Fs00205-024-01984-y/MediaObjects/205_2024_1984_Equ371_HTML.png)
Thus, integrating by parts and using that \(\nabla u= - \nu \) on \(\partial \Omega _u\cap A_R\), we obtain
![](http://media.springernature.com/lw504/springer-static/image/art%3A10.1007%2Fs00205-024-01984-y/MediaObjects/205_2024_1984_Equ372_HTML.png)
where H is the mean curvature of \(\partial \Omega _u\cap A_R\) oriented towards the complement of \(\Omega _u\).
Step 4. Conclusion. Given any \(\varphi \in C^\infty _c(\mathbb {R}^d{\setminus }\{0\})\), we consider an annulus \(A_R\) containing the support of \(\varphi \) and the vector field \(\xi :=\varphi \nabla u\). Thus, by the minimality of \(u'\) in \(\Omega _u\cap A_R\), we get
where the last inequality follows from (7.11). \(\square \)
8 Hausdorff Dimension of \(\text { Sing}(\partial \Omega )\)
In this section we will estimate the dimension of the singular set \(\text { Sing}(\partial \Omega )\) of a solution \(\Omega \) to the shape optimization problem (1.3); our main result is the following.
Theorem 8.1
(Dimension of the singular set) Let D be a bounded open set in \(\mathbb {R}^d\) and let f, g, Q be as in Theorem 1.2, namely:
-
(a)
\(f,g\in C^2(D)\cap L^\infty (D)\);
-
(b)
there are constants \(C_1,C_2 > 0\) such that \(0\le C_1 g\le f\le C_2 g\) in D;
-
(c)
\(Q\in C^{2}(D)\) and there are constants \(c_Q,C_Q\) such that \(0<c_Q\le Q\leqq C_Q\) on D.
Let \(\Omega \) be a solution to (1.3) and let the singular part \(\text { Sing}(\partial \Omega )\) of the free boundary \(\partial \Omega \cap D\) be as in Sect. 5. Then, the following holds:
-
(i)
If \(d< d^*\), then \(\text { Sing}(\partial \Omega )=\emptyset \).
-
(ii)
If \(d\ge d^*\), then the Hausdorff dimension of \(\text { Sing}(\partial \Omega )\) is at most \(d-d^*\), that is,
$$\begin{aligned} \mathcal {H}^{d-d^*+{\varepsilon }}\big (\text { Sing}(\partial \Omega )\big )=0\quad \text {for every}\quad {\varepsilon }>0. \end{aligned}$$
In order to prove Theorem 8.1 above, we will first show that the stability of \(\Omega \), expressed in terms of the state functions \(u_\Omega \) and \(v_\Omega \) as in Lemma 2.8, passes to a blow-up limit.
Lemma 8.2
(Stability of the blow-up limits) Let D be a bounded open set in \(\mathbb {R}^d\) and let f, g, Q be as in Theorems 1.2 and 8.1. Let \(\Omega \) be an optimal set for (1.3) and let \(u:=u_\Omega \) and \(v:=v_\Omega \) be the state functions defined in (1.1) and (1.6). Suppose that the couple \(u_0,v_0:\mathbb {R}^d\rightarrow \mathbb {R}\) is a blow-up limit of u, v at a point \(x_0\in \partial \Omega \cap D\). Then,
where \(\Omega _0:=\{u_0>0\}=\{v_0>0\}\), and \(\delta u_0\) and \(\delta v_0\) are the solutions respectively to the PDEs
in the sense explained in Sect. 7.1. In particular, if \(u_0\) and \(v_0\) are proportional, then there is a constant \(\lambda >0\) such that \(\lambda u_0\) is a global stable solution of the one-phase Bernoulli problem in the sense of Definition 7.3.
Proof
Let \(r_k\rightarrow 0\), be a sequence such that
converge respectively to \(u_0\) and \(v_0\) locally uniformly and (by Proposition 4.3) strongly in \(H^1_{loc}(\mathbb {R}^d)\). We define:
Then, the functions \(u_k\) and \(v_k\) satisfy the PDEs
where
Moreover, \(\Omega _k\) is optimal in the rescaled domain
for the functional
where, this time, by \(u_A\) and \(v_A\) we denote the solutions to
We fix a compactly supported smooth vector field \(\xi \in C^\infty _c(\mathbb {R}^d;\mathbb {R}^d)\). Since \(r_k\rightarrow 0\), for k large enough the support of \(\xi \) is contained in \(D_k\) and the stability of \(\Omega _k\) (Lemma 2.8) reads as
where \(\delta u_k\) and \(\delta v_{k}\) are the solutions to
and where we used the following notation:
-
\(\delta A\) and \(\delta ^2A\) are the matrices defined in (2.6) (we notice that \(\delta A\) and \(\delta ^2A\) are defined in terms of \(\xi \) only);
-
the variations \(\delta f_{k}\) and \(\delta ^2 f_{k}\) (\(\delta g_{k}\) and \(\delta ^2 g_{k}\) are defined analogously) are given by;
$$\begin{aligned} \begin{aligned} \delta f_k(x)&:={{\,\textrm{div}\,}}(f_k\xi )=r_k^2\nabla f(x_0+r_kx)\cdot \xi +r_kf(x_0+r_kx)\,{{\,\textrm{div}\,}}\,\xi \,,\\ \delta ^2 f_k(x)&:= \frac{r_k^3}{2} \xi \cdot (D^2f(x_0+r_kx))\xi +\frac{r_k^2}{2}\nabla f(x_0+r_kx)\cdot D\xi [\xi ]\\&\quad +r_kf(x_0+r_kx)\frac{({{\,\textrm{div}\,}}\xi )^2+\xi \cdot \nabla [{{\,\textrm{div}\,}}\,\xi ]}{2}+r_k^2(\nabla f(x_0+r_kx) \cdot \xi ){{\,\textrm{div}\,}}\xi \,, \end{aligned} \end{aligned}$$ -
\(\delta ^2Q_k\) is given by
$$\begin{aligned} \begin{aligned} \delta ^2Q_k(x)&:=(\xi \cdot \nabla Q_k){{\,\textrm{div}\,}}\xi +\frac{1}{2}\xi \cdot D^2 Q_k \xi +Q_k\frac{1}{2}\Big (({{\,\textrm{div}\,}}\xi )^2+\xi \cdot \nabla ({{\,\textrm{div}\,}}\xi )\Big )\\&=r_k(\xi \cdot \nabla Q(x+r_kx)){{\,\textrm{div}\,}}\xi +\frac{r_k^2}{2}\xi \cdot D^2 Q(x_0+r_kx) \xi \\&\qquad +\frac{1}{2}Q(x_0+r_k)\Big (({{\,\textrm{div}\,}}\xi )^2+\xi \cdot \nabla ({{\,\textrm{div}\,}}\xi )\Big ), \end{aligned} \end{aligned}$$where in all the formulas above, the field \(\xi \) and its derivatives are all computed in x.
We notice that \(\delta f_k\) and \(\delta ^2f_k\) vanish outside the support of \(\xi \). Moreover, since f and g are \(C^2\), we get that \(\delta f_k\) and \(\delta ^2f_k\) converge to zero uniformly. Similarly, since Q is \(C^2\), we get that
Next, we notice that by Proposition 4.3, \(u_k\) and \(v_k\) converge strongly in \(H^1_{loc}(\mathbb {R}^d)\) to the blow-up limits \(u_0\), \(v_0\). Thus, since the support of \(\delta ^2A\) is compact, we get
Similarly, we have that
strongly in \(L^2(\mathbb {R}^d;\mathbb {R}^d)\). Thus, by Lemma 7.2, we get that \(\delta u_k\) and \(\delta v_k\) converge respectively to the solutions \(\delta u_0\) and \(\delta v_0\) of (8.2). Now, (8.1) follows by passing to the limit (8.3). \(\square \)
Proof of Theorem 8.1
The strategy is similar to the one for minimizers of the one-phase problem. We split the proof in two different cases.
Case 1: \(d<d^*\). Let \(x_0 \in \partial \Omega _u\cap D\) and \(r_k \rightarrow 0^+\) be the infinitesimal sequence such that the rescalings \(u_{x_0,r_k}\) and \(v_{x_0,r_k}\) converge to the 1-homogeneous blow-up limits \(u_0\) and \(v_0\) given by Proposition 4.7. Then, given \(\lambda \in (C_1,C_2)\) as in Proposition 4.7, the blow-up limits
coincide up to a multiplicative constant (that is \(v_0=Q(x_0)v_0\)) and, by Lemma 8.2 and Proposition 4.7, \(u_0\) is a 1-homogeneous stable solutions of the one-phase problem in the sense of Definition 7.3. Therefore, by definition of \(d^*\), \(\text { Sing}(\partial \Omega _{u_0})=\emptyset \) and \(u_0\) is a half space solution:
Finally, by rewriting the result in terms of the blow-up sequence of Proposition 4.7, we deduce the existence \(\alpha >0\) and \(\beta >0\) such that \(\alpha \beta =Q(x_0)\) such that
as \(k\rightarrow \infty \). Thus, by definition, \(x_0 \in \text { Reg}(\partial \Omega )\). Since this is true at every free boundary point \(x_0\in \partial \Omega \cap D\), we get that \(\text { Sing}(\partial \Omega )=\emptyset \).
Case 2: \(d\geqq d^*\). Given \({\varepsilon }>0\), let us prove that
We will apply consecutively the three blow-ups from Sect. 4. By contradiction, assume that
Therefore, by [37, Lemma 10.5], there are \(x_0 \in \text { Sing}(\partial \Omega )\) and a sequence \(r_k\rightarrow 0\) such that
By the first blow-up analysis from Sect. 4.2 (see Proposition 4.3), we deduce that the blow-up sequences \(u_{x_0,r_k},v_{x_0,r_k}\) and \(\Omega _{x_0,r_k}\) converge to some limits \(u_0, v_0\) and \(\Omega _0\) such that
where \(u_0\), \(v_0\) and \(\Omega _0\) are as in Proposition 4.3 and where \(u_0\) and \(v_0\) satisfy the stability condition (8.1) from Lemma 8.2. By applying again [37, Lemma 10.5], there exists a point \(x_{00} \in \text { Sing}(\partial \Omega _0)\) and another sequence \(r_k\rightarrow 0^+\) such that
Hence, by applying the second blow-up analysis of Sect. 4.3 to the sequences
we get that
where \(v_{00}=Q(x_0) u_{00}\) and \(u_{00}\) satisfies (2)-(3)-(4)-(5)-(6) in Lemma 4.4. Moreover, by the strong convergence of the blow-up sequence, \(u_{00}\) and \(v_{00}\) still satisfy (8.1), so \(u_{00}\) is a global stable solution of the one-phase problem in the sense of Definition 7.3. Finally, by applying for the last time [37, Lemma 10.5], we deduce that
in which \(\Omega _{000}=\{u_{000}>0\}\) and \(u_{000}\) is a blow-up limit of \(u_{00}\) at some free boundary point \(x_{000}\in \partial \Omega _{00}\). But now \(u_{000}\) is a 1-homogeneous stable solution of the one-phase problem in the sense of Definition 7.3, so (8.5) contradicts Theorem 7.8. \(\square \)
9 Existence of Optimal Sets in \(\mathbb {R}^d\) and Proof of Theorem 1.6
In this section we prove Theorem 1.6; we prove the existence of an optimal set for (1.4) and then we show how to obtain the regularity of the optimal sets as a consequence from Theorem 1.2.
9.1 Statement of the Problem in the Class of Measurable Sets
One can not obtain the existence of optimal sets (for (1.4) or (1.3)) directly in the class of open sets. Thus, we extend the definition of the shape functional to the class of measurable sets. Precisely, if \(\Omega \) is a Lebesgue measurable set of finite measure in \(\mathbb {R}^d\), we define the space \({\widetilde{H}}^1_0(\Omega )\) of all \(H^1(\mathbb {R}^d)\) functions vanishing (Lebesgue-)almost-everywhere outside \(\Omega \). We will say that u is a (weak) solution to the problem
if \(u\in {\widetilde{H}}^1_0(\Omega )\) and
It is immediate to check that u satisfies (9.2) if and only if
From now on, we will denote the unique solution to (9.1) by \({\widetilde{u}}_\Omega \).
We will first prove the following lemma, which in particular implies that for every open set \(\Omega \subset \mathbb {R}^d\) of finite measure and non-negative functions \(f,g\in L^2(\Omega )\), we have
where \(u_\Omega \) is the weak solution to (1.1).
Lemma 9.1
Let \(\Omega \) be a bounded open set in \(\mathbb {R}^d\) and \(f\in L^2(\Omega )\) a non-negative function. Then
where \({\widetilde{u}}_\Omega \in {\widetilde{H}}^1_0(\Omega )\) is the solution to (9.1) and \(u_\Omega \in H^1_0(\Omega )\) is the weak solution to (1.1).
Moreover, if \(\Omega \) satisfies the following exterior density estimate:
then \(H^1_0(\Omega )={\widetilde{H}}^1_0(\Omega )\) and \( u_\Omega ={\widetilde{u}}_\Omega \).
Proof
First we notice that \(u_\Omega \) and \({\widetilde{u}}_\Omega \) are the unique minimizers of the functional (9.3) in \(H^1_0(\Omega )\) and \({\widetilde{H}}^1_0(\Omega )\). Since \(f\ge 0\), by testing the optimality of \(u_\Omega \) with \(u_\Omega \vee 0\in H^1_0(\Omega )\) and the optimality of \({\widetilde{u}}_\Omega \) with \({\widetilde{u}}_\Omega \vee 0\in \widetilde{H}^1_0(\Omega )\), we get that \(u_\Omega \ge 0\) and \({\widetilde{u}}_\Omega \ge 0\) in \(\Omega \). Now, a standard argument (see for instance [38, Lemma 2.6]) gives that \(\Delta u_\Omega +f\ge 0\) on \(\mathbb {R}^d\) in the sense of distributions, precisely:
Next, we notice that for every open set \(\Omega \), we have the inclusion \(H^1_0(\Omega )\subset {\widetilde{H}}^1_0(\Omega )\). Thus, if we take \(\varphi \) to be the negative part of \({\widetilde{u}}_\Omega -u_\Omega \),
we have that \(\varphi \in {\widetilde{H}}^1_0(\Omega )\). So, using the (weak) equation for \({\widetilde{u}}_\Omega \), we get
On the other hand, by the positivity of \(\varphi \) and (9.6),
Combining the two, we obtain
which gives that \(\varphi \equiv 0\) and so (9.4) holds. Finally, it is known (see for instance [17, Proposition 4.7]) that in the presence of the density estimate (9.5), we have that \(H^1_0(\Omega )={\widetilde{H}}^1_0(\Omega )\). Thus, the equality \(u_\Omega ={\widetilde{u}}_\Omega \) follows by the fact that the minimizer of the functional in (9.3) is unique. \(\square \)
9.2 Existence of Optimal Measurable Sets in \(\mathbb {R}^d\)
For all measurable set \(\Omega \subset \mathbb {R}^d\), we set
We can prove the following existence result for the minimization of \({\widetilde{J}}\).
Lemma 9.2
If \(d\ge 3\), let \(f,g\in L^2(\mathbb {R}^d)\), while if \(d=2\), let \(f,g\in L^1(\mathbb {R}^2)\cap L^\infty (\mathbb {R}^2)\) be nonnegative functions, and let \(Q:\mathbb {R}^d\rightarrow \mathbb {R}\) be a measurable function bounded from below by a positive constant \(c_Q>0\). Then, the following variational problem has a solution
Proof
Let \(d\ge 3\), let \(\Omega _n\) be a minimizing sequence of sets of finite Lebesgue measure for (9.7). We set \(u_n:={\widetilde{u}}_{\Omega _n}\). By the Poincaré inequality, the equation for \(u_n\) and the Hölder inequality, there is a dimensional constant \(C_d\) such that
Thus,
and so (since we can suppose that \({\widetilde{J}}(\Omega _n)\le {\widetilde{J}}(\emptyset )=0\)), we get
which implies that the sequence of measures \(|\Omega _n|\) is bounded
If \(d=2\), to obtain a bound on the measure analogous to (9.8), we need to use a generalized Poincaré-Sobolev (or generalized Faber–Krahn) inequality (see for example [3, equation (1.2)]) instead of the classical one, namely, for \(2<q<+\infty \), there exists a dimensional constant \({\widetilde{C}}_{2}\) such that (using also Hölder inequality)
We immediately deduce that
and again supposing that \({\widetilde{J}}(\Omega _n)\le \widetilde{J}(\emptyset )=0\), we obtain, using (9.9)
which, since \(q>2\) implies that the sequence of measures \(|\Omega _n|\) is bounded, namely
Now the proof continues in the same way both for \(d=2\) and \(d\geqq 3\). Using again the equation for \(u_n\) we get that the sequence \(u_n\) is bounded in \(H^1(\mathbb {R}^d)\). Thus, up to pass to a subsequence, we have that \(u_n\) converges weakly in \(H^1_{\textrm{loc}}(\mathbb {R}^d)\) and strongly in \(L^2_{\textrm{loc}}(\mathbb {R}^d)\) (and, up to another subsequence, also pointwise a.e. in \(\mathbb {R}^d\)) to a certain \(u\in H^1_{\textrm{loc}}(\mathbb {R}^d).\) Now, for all \(R>0\)
Taking a supremum over \(R>0\), we obtain that
and as a consequence (using Fatou Lemma),
Then one can prove also that \(u\in H^1(\mathbb {R}^d)\cap \widetilde{H}^1_0(\{u\ne 0\})\), as, for all \(R>0\),
and then we can take the supremum over \(R>0\). We note that from the pointwise a.e. convergence and the fact that \(u_n\geqq 0\) (since \(f\geqq 0\)), we deduce \(u\geqq 0\). As a consequence, using also that \(g\geqq 0\), for all \(R>0\) we have
We then conclude that \(\Omega =\{u\ne 0\}\) is a solution to (9.7). \(\square \)
Lemma 9.3
Let \(d\ge 2\), \(f,g\in L^1(\mathbb {R}^d)\cap L^\infty (\mathbb {R}^d)\) be non-negative, and let \(Q:\mathbb {R}^d\rightarrow \mathbb {R}\) be a measurable function bounded from below by a positive constant \(c_Q>0\). If the measurable set \(\Omega \subset \mathbb {R}^d\), \(|\Omega |<+\infty \), is a solution to (9.7), then \(\Omega \) is bounded.
Proof
For any set \(E\subset \mathbb {R}^d\) of finite measure and any function \(h\in L^2(E)\), we will denote by \({\widetilde{R}}_E(h)\) the unique solution to the problem
We have that \({\widetilde{R}}_E\) is linear, \(\widetilde{R}_E(h_1+h_2)={\widetilde{R}}_E(h_1)+{\widetilde{R}}_E(h_2)\) and positive: if \(h\ge 0\), then \({\widetilde{R}}_E(h)\ge 0\). Moreover, by the weak maximum principle, if \(E_1\subset E_2\) and \(h\ge 0\), then \({\widetilde{R}}_{E_2}(h)\ge {\widetilde{R}}_{E_1}(h).\)
Let \(\omega \) be any measurable set contained in \(\Omega \). Then, the minimality of \(\Omega \) gives that
So, rearranging the terms and using the positivity of f and g, and the inequality
we get
Finally, rearranging the terms again, we get that
so the set \(\Omega \) is inwards minimizing (or, in terms of [9], a shape subsolution) for the functional
Thus, applying [9, Theorem 3.13], for every
the set \(\Omega \) is contained in an open set \(A\subset \mathbb {R}^d\) obtained as a finite union of N balls \(B_\rho (x_i)\), \(i=1,\dots ,N\), with N and \(\rho \) depending only on the dimension d and \(\eta \). In particular, A is bounded and the diameter of any connected component of A is at most \(N\rho \). \(\square \)
9.3 Proof of Theorem 1.6
The existence of an optimal set is a consequence of Proposition 9.4 below, while the regularity follows from Theorem 1.2.
Proposition 9.4
In \(\mathbb {R}^d\), \(d\ge 2\), let \(f,g,Q:\mathbb {R}^d\rightarrow \mathbb {R}\) be non-negative functions. Suppose that:
-
(a)
\(f,g\in L^\infty (\mathbb {R}^d)\cap L^1(\mathbb {R}^d)\cap C(\mathbb {R}^d)\) and that \(f>0\) and \(g>0\) on \(\mathbb {R}^d\) ;
-
(b)
there is a constant \(c_Q>0\) such that \(c_Q\le Q\) on \(\mathbb {R}^d\).
Then, there exists a solution \(\Omega \subset \mathbb {R}^d\) to the shape optimization problem (1.4). Moreover, every solution to (1.4) is also a solution to (9.7).
Proof
By Lemma 9.2, there is a measurable set \(\Omega \) (of finite measure) that minimizes (9.7). By Lemma 9.3, \(\Omega \) is contained in some ball \(B_R\subset \mathbb {R}^d\). Since f and g are continuous and strictly positive on \({\overline{B}}_R\), we can find positive constants \(C_1,C_2\) such that \(C_1g\le f\le C_2g\) on \({\overline{B}}_R\). Thus, reasoning as in Proposition 3.1 and Corollary 3.3 we get that the set
is open. Now, the optimality of \(\Omega \) gives that \(|\Omega \Delta A|=0\), so we have \({\widetilde{u}}_A={\widetilde{u}}_\Omega .\) Moreover, by Proposition 3.5, A satisfies the exterior density estimate from Lemma 9.1. Thus, \(u_A={\widetilde{u}}_A\). In order to check that the open set A minimizes (1.4), we notice that, for any open set \(E\subset \mathbb {R}^d\),
Moreover, by the same chain of inequalities, we obtain that if the open set E is a solution to (1.4), then it also minimizes (9.7). \(\square \)
Data Availability Statement
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Notes
Precisely, since \(\mathcal G(u)=\infty \) as soon as u has positivity set of infinite measure, we will work with a localized version of \(\mathcal G\), in which the integration is over compact sets.
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Acknowledgements
G.T. and B.V. are supported by the European Research Council’s (ERC) project n.853404 ERC VaReg - Variational approach to the regularity of the free boundaries, financed by the program Horizon 2020. G.B., F.P.M., D.M. and G.T. are members of INDAM-GNAMPA. D.M. has been partially supported by the MIUR-PRIN Grant 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)”. D.M. and B.V. have been partially supported by the MIUR-PRIN Grant 2022R537CS “\(NO^3\)” granted by the European Union - Next Generation EU. D.M. and G.T. have been partially supported by the INDAM-GNAMPA project CUP_E53C22001930001.
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Appendix A. An Optimization Problem in Heat Conduction
Appendix A. An Optimization Problem in Heat Conduction
As a consequence of our analysis from Sects. 2, 4 and 7, we obtain an estimate on the dimension of the singular part of the boundary of the optimal sets arising in a heat conduction problem studied by Aguilera, Caffarelli and Spruck in [1].
1.1 A.1 Statement of the Problem and Known Results
Let D be a smooth bounded open set in \(\mathbb {R}^d\). Let \(\phi :\partial D\rightarrow \mathbb {R}\) be a smooth function on \(\partial D\) bounded from below and above by positive constants \(0<c\le C\). For every open set \(\Omega \subset D\), for which the set \(K:= D\setminus \Omega \) is a compact subset of D, we consider the state function \(u_\Omega \in H^1(D)\) solution to the problem
and we define the functional
where \(\Lambda >0\) is a real number and \(\nu \) is the outer unit normal to \(\partial D\). In [1] it was shown that there is a solution to the following shape optimization problem:
and that for every solution \(\Omega \) to (A.2) the following holds.
-
The state function \(u_\Omega \) is Lipschitz continuous in D and smooth in a neighborhood of the fixed boundary \(\partial D\).
-
\(u_\Omega \) is non-degenerate in the sense that there is a constant \(\eta >0\), for which
$$\begin{aligned} \frac{1}{r^{d-1}}\int _{\partial B_r(x)}u_\Omega \,d\mathcal {H}^{d-1}\ge \eta \,r\quad \text {whenever}\quad x\in {\overline{\Omega }}\quad \text {and}\quad B_r(x)\subset D. \end{aligned}$$ -
There is \({\varepsilon }>0\) such that, for every x on the boundary of the set \(K:=D\setminus \Omega \), we have:
$$\begin{aligned} {\varepsilon }_0|B_r|\le |\Omega \cap B_r(x)|\le (1-{\varepsilon }_0)|B_r|\quad \text {whenever}\quad B_r(x)\subset D. \end{aligned}$$ -
There is a constant \(\tilde{M}>0\) such that, for every \(B_{2r}(x_0)\subset D\), we have the bound
$$\begin{aligned} 0\le \bigg |\int _{B_{2r}(x_0)}\nabla u_\Omega \cdot \nabla \varphi \,dx\bigg |\le \tilde{M} \Vert \varphi \Vert _{L^\infty (B_{2r}(x_0))}\quad \text {for every}\quad \varphi \in C^{0,1}_c(B_{2r}(x_0)), \end{aligned}$$where \(C^{0,1}_c(B_{2r}(x_0))\) is the space of Lipschitz functions with compact support in \(B_{2r}(x_0)\).
Then, they showed that the reduced boundary \(\partial ^*\Omega \cap D\) is \(C^\infty \) smooth and that
where \(v_\Omega \) is the solution to the PDE
In Theorem A.4, we will improve this result in low dimension (\(d\ge 4\)) by showing that the whole free boundary \(\partial \Omega \cap D\) is smooth.
Remark A.1
In the problem originally considered by Aguilera, Caffarelli and Spruck in [1], the minimization of \(\mathcal {F}\) is among all domains with prescribed measure. Thus, all the results from [1] apply to (A.2) with the only difference that the constant \(\Lambda \) from (A.3) is an unknown positive Lagrange multiplier. In this last section of the present paper, we choose to work with the penalized version (A.2) since it allows to apply all the results from Sects. 2, 7 and 8 directly, without the need to add technical details related to the Lagrange multiplier.
1.2 A.2 First and Second Variation
By an integration by parts, we can rewrite \(\mathcal {F}(\Omega )\) as
where \(u_\Omega \) and \(v_\Omega \) are given by (A.1) and (A.4). Now, let \(\xi \in C^\infty _c(D;\mathbb {R}^d)\) be a smooth compactly supported vector field in D; let \(\Phi _t:D\rightarrow D\), \(t\in \mathbb {R}\), be the flow associated to \(\xi \) and let \(\Omega _t:=\Phi _t(\Omega )\). Reasoning as in Sect. 7, we get that
where \(\delta A\) and \(\delta ^2A\) are given by (2.6), and where \(\delta u\) and \(\delta v\) are the solutions to the PDEs
Using the blow-up analysis from Sect. 4 and the argument from Lemma 8.2, we get
Lemma A.2
Let \(\Omega \) be an optimal set for (A.2) and let \(u:=u_\Omega \) and \(v:=v_\Omega \) be the state functions from (A.1) and (A.4). Suppose that \(x_0\in \partial \Omega \cap D\) and that \(r_k\rightarrow 0\) are such that
where \(u_0,v_0\) are non-negative Lipschitz functions on \(\mathbb {R}^d\) and both limits are locally uniform in \(\mathbb {R}^d\), and where, as usual, we set:
Then, \(u_0\) and \(v_0\) are proportional and there is a constant \(\lambda >0\) such that \(\lambda u_0\) is a global stable solution of the one-phase Bernoulli problem in the sense of Definition 7.3.
Proof
By the analysis of the blow-up sequences from Sect. 4, we get that \(u_{x_0,r_k}\) and \(v_{x_0,r_k}\) converge strongly in \(H^1_{loc}(\mathbb {R}^d)\) respectively to \(u_0\) and \(v_0\), and also that the sets \(\frac{1}{r_k}(\Omega -x_0)\) converge (in the local Hausdorff distance) to \(\Omega _0=\{u_0>0\}=\{v_0>0\}\). Then, as in Lemma 8.2, for every smooth compactly supported vector field \(\xi \in C^\infty _c(\mathbb {R}^d;\mathbb {R}^d)\), the first and the second variation of \(\mathcal {F}\) pass to the limit, that is,
where \(\delta A\) and \(\delta ^2A\) are as in (2.6), and where \(\delta u_0\) and \(\delta v_0\) solve the PDE (see Sect. 7.1)
On the other hand, since by [1] the Boundary Harnack Principle holds on the optimal set \(\Omega \), we get that the blow-up limits are proportional, that is, \(v_0=Cu_0\) for some \(C>0\). Finally, we notice that, up to multiplying \(u_0\) by a constant, the inequalities (A.5) correspond precisely to the stability condition (7.8) in Definition 7.3, which concludes the proof of the lemma. \(\square \)
1.3 A.3 Main Theorem
Let now \(\Omega \) be an optimal set for (A.2). We define the regular and the singular parts of the free boundary \(\partial \Omega \cap D\) as follows. The regular part, \(\text { Reg}(\partial \Omega )\), is the set of points \(x_0\in \partial \Omega \cap D\) at which there exists a blow up limit \(u_0,v_0:\mathbb {R}^d\rightarrow \mathbb {R}\) of the form
for some unit vector \(\nu \in \mathbb {R}^d\) and some constants \(\alpha >0\) and \(\beta >0\) such that \(\alpha \beta =\Lambda \). The remaining part of the free boundary is the singular set:
Then, in terms of the decomposition into \(\partial \Omega \cap D=\text { Reg}(\partial \Omega )\cup \text { Sing}(\partial \Omega )\), we can rewrite the regularity theorem of Aguilera–Caffarelli–Spruck (for the penalized problem) as follows:
Theorem A.3
(Aguilera–Caffarelli–Spruck [1]) If \(\Omega \) is a solution to (A.2) in a domain \(D\subset \mathbb {R}^d\), then \(\text { Reg}(\partial \Omega )\) is a relatively open subset of \(\partial \Omega \cap D\) and (locally) a \(C^\infty \) manifold.
For what concerns the singular set, by reasoning as in Theorem 8.1 and by applying Theorems 7.9 and 7.8, we obtain the following result, which in particular implies that in the physically relevant dimension \(d=3\) (and also in \(d=2\) and \(d=4\)) the free boundary \(\partial \Omega \cap D\) of a solution \(\Omega \) to (A.2) is \(C^\infty \) smooth.
Theorem A.4
(Dimension of the singular set for solutions to (A.2)) If \(\Omega \) is a solution to (A.2) in a domain \(D\subset \mathbb {R}^d\), then the following holds:
-
(i)
if \(d< d^*\), then \(\text { Sing}(\partial \Omega )=\emptyset \);
-
(ii)
if \(d\ge d^*\), then the Hausdorff dimension of \(\text { Sing}(\partial \Omega )\) is at most \(d-d^*\);
where \(d^*\) is the critical dimension from Definition 7.7.
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Buttazzo, G., Maiale, F.P., Mazzoleni, D. et al. Regularity of the Optimal Sets for a Class of Integral Shape Functionals. Arch Rational Mech Anal 248, 44 (2024). https://doi.org/10.1007/s00205-024-01984-y
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DOI: https://doi.org/10.1007/s00205-024-01984-y