A free boundary problem for the p-Laplacian with nonlinear boundary conditions

We study a nonlinear generalization of a free boundary problem that arises in the context of thermal insulation. We consider two open sets $\Omega\subseteq A$, and we search for an optimal $A$ in order to minimize a non-linear energy functional, whose minimizers $u$ satisfy the following conditions: $\Delta_p u=0$ inside $A\setminus\Omega$, $u=1$ in $\Omega$, and a nonlinear Robin-like boundary $(p,q)$-condition on the free boundary $\partial A$. We study the variational formulation of the problem in SBV, and we prove that, under suitable conditions on the exponents $p$ and $q$, a minimizer exists and its jump set satisfies uniform density estimates.


Introduction
Let Ω ⊆ R n be a bounded open set with smooth boundary, and let A be a set containing Ω.

Consider the functional
with v ∈ H 1 (A), v = 1 in Ω and β, C 0 > 0 fixed positive constants.The problem of minimizing this functional arises in the environment of thermal insulation: F represents the energy of a heat configuration v when the temperature is maintained constant inside the body Ω and there's a bulk layer A \ Ω of insulating material whose cost is represented by C 0 and the heat transfer with the external environment is conveyed by convection.For simplicity's sake in the following we will set C 0 = 1.The variational formulation in (1.1) leads to an Euler-Lagrange equation, which is the weak form of the following problem: The problems we are interested in concern the existence of a solution and its regularity.In this sense, one could be interested in studying a more general setting in which it is possible to consider possibly irregular sets A. Specifically, we could generalize the problem into the context of SBV functions, aiming to minimize the functional with v ∈ SBV(R n ) and v = 1 in Ω.This problem has been studied by L. A. Caffarelli and D. Kriventsov in [5], where the authors have proved the existence of a solution u for the problem and the regularity of its jump set.Another similar problem, in a non-linear context, has been deepened by D. Bucur and A. Giacomini in [4] with a boundedness constraint.
In this paper, our main aim is to generalize the problem and techniques employed in [5] to a nonlinear formulation.In detail, for p, q > 1 fixed, we consider the functional and in the following we are going to study the problem Notice that if v ∈ SBV(R n ) with v = 1 a.e. in Ω, letting v 0 = max { 0, min { v, 1 } } we have that v 0 ∈ SBV(R n ) with v 0 = 1 a.e. in Ω and F(v 0 ) ≤ F(v) so it suffices to consider the problem In a more regular setting, problem (1.4) can be seen as a PDE.Let us fix Ω, A sufficiently smooth open sets, u ∈ W 1,p (A) with u = 1 on Ω, and let us define the functional minimizers u to (1.5) solve the following boundary value problem (1.6) In Section 2 we give some preliminary tools and definitions, and then we will prove the existence of a minimizer u of (1.4), under a prescribed condition on p and q.Finally, we will prove density estimates for the jump set J u .
We resume in the following theorems the main results of this paper.
Then there exists a solution u to problem (1.4) and there exists a constant δ 0 = δ 0 (Ω, β, p, q) > 0 such that u > δ 0 (1.7) L n -almost everywhere in { u > 0 }, and there exists ρ(δ 0 ) > 0 such that Let Ω ⊆ R n be a bounded open set, and let p, q > 1 be exponents satisfying the assumptions of Theorem 1.1.Then there exist positive constants C(Ω, β, p, q), c(Ω, β, p, q), C 1 (Ω, β, p, q) such that if u is a minimizer to problem (1.4), then and In particular, this implies the essential closedness of the jump set J u , namely In Section 3 we prove that the a priori estimate (1.7) holds for inward minimizers (see Definition 3.1), such an estimate will be crucial in the proof of Theorem 1.1 in Section 4. Finally, in Section 5 we prove Theorem 1.2.
Remark 1.3.Notice that the condition on the exponents is undoubtedly verified when p ≥ q > 1.Furthermore, if Ω is a set with Lipschitz boundary, the exponent p * is the optimal exponent such that

Notation and Tools
In this section, we give the definition of the space SBV, and some useful notations and results that we will use in the following sections.We refer to [1], [6], [2] for a more intensive study of these topics.
We say that u is a function of bounded variation in R n and we write u ∈ BV(R n ) if its distributional derivative is a Radon measure, namely with Du a R n -valued measure in R n .We denote with |Du| the total variation of the measure Du.The space BV(R n ) is a Banach space equipped with the norm Definition 2.2.Let E ⊆ R n be a measurable set.We define the set of points of density 1 for E as and the set of points of density 0 for E as Moreover, we define the essential boundary of E as ).
Definition 2.3 (Approximate upper and lower limits).Let u : R n → R be a measurable function.We define the approximate upper and lower limits of u, respectively, as We define the jump set of u as We denote by K u the closure of J u .
If u(x) = u(x) = l, we say that l is the approximate limit of u as y tends to x, and we have that, for any ε > 0, lim sup If u ∈ BV(R n ), the jump set J u is a (n − 1)-rectifiable set, i.e.J u ⊆ i∈N M i , up to a H n−1 -negligible set, with M i a C 1 -hypersurface in R n for every i.We can then define H n−1 -almost everywhere on J u a normal ν u coinciding with the normal to the hypersurfaces M i .Furthermore, the direction of ν u (x) is chosen in such a way that the approximate upper and lower limits of u coincide with the approximate limit of u on the half-planes Definition 2.4.Let Ω ⊆ R n be an open set, and E ⊆ R n a measurable set.We define the relative perimeter of E inside Ω as If P (E; R n ) < +∞ we say that E is a set of finite perimeter.
Theorem 2.5 (Decomposition of BV functions).Let u ∈ BV(R n ).Then we have where ∇u is the density of Du with respect to the Lebesgue measure, ν u is the normal to the jump set J u and D c u is the Cantor part of the measure Du.The measure D c u is singular with respect to the Lebesgue measure and concentrated out of J u .
Definition 2.6.Let v ∈ BV(R n ), let Γ ⊆ R n be a H n−1 -rectifiable set, and let ν(x) be the generalized normal to Γ defined for H n−1 -a.e.x ∈ Γ.For H n−1 -a.e.x ∈ Γ we define the traces γ ± Γ (v)(x) of v on Γ by the following Lebesgue-type limit quotient relation and they coincide with the approximate limit of v in x.In particular, if Γ = J v , we have We now focus our attention on the BV functions whose Cantor parts vanish.Definition 2.8 (SBV).Let u ∈ BV(R n ).We say that u is a special function of bounded variation and we write u For SBV functions we have the following.
Theorem 2.9 (Chain rule).Let g : R → R be a differentiable function.
Furthermore, if g is increasing, while, if g is decreasing, We now state a compactness theorem in SBV that will be useful in the following.
Then there exists u ∈ SBV(R n ) and a subsequence u k j such that • Compactness: • Lower semicontinuity: for every open set A we have We refer to [1, Theorem 4.7, Theorem 4.8, Theorem 5.22] for the proof of this theorem.We now conclude this section with the following proposition whose proof can be found in [5,Lemma 3.1].

Lower Bound
In the following, we assume that Ω ⊂ R n is a bounded open set and that p and q are two positive real numbers such that where p ′ and q ′ are the Hölder conjugates of p and q respectively.
for every set of finite perimeter A containing Ω, where χ A is the characteristic function of set A.
Let A ⊂ R n be a set of finite perimeter such that Ω ⊂ A, and let v ∈ SBV(R n ).We will make use of the following expression Let B be a ball containing Ω, then χ B ∈ SBV(R n ) and χ B = 1 in Ω, we will denote F(χ B ) by F.
Theorem 3.2.There exists a positive constant δ = δ(Ω, β, p, q) such that if u is an inward minimizer with F(u) ≤ 2 F , then u > δ Proof.Let 0 < t < 1 and For every such t, we have and rearranging the terms, On the other hand, , where we used and γ > 1 by (3.1).By classical BV embedding in L 1 * applied to the function (uχ { u≤t } ) q and the estimate (3.4), we have We can compute We therefore get Let 0 < t 0 < 1 such that f (t 0 ) > 0, then for every t 0 < t < 1, we have f (t) > 0 and integrating from t 0 to 1, we have for every t 0 < δ we would have f (t 0 ) < 0, which is a contradiction.Therefore f (t) = 0 for every t < δ, from which u > δ L n -almost everywhere on { u > 0 }.
Proposition 3.4.There exists a positive constant δ 0 = δ 0 (Ω, β, p, q) < δ such that if u is an inward minimizer with F(u) ≤ 2 F , then u is supported on B ρ(δ 0 ) , where ρ(δ 0 ) = δ 1−q 0 and B ρ(δ 0 ) is the ball centered at the origin with radius ρ(δ 0 ).Moreover there exist positive constants C(Ω, β, p, q), C 1 (Ω, β, p, q) such that, for any B r (x) ⊆ R n \ Ω we have ) Proof.By Theorem 3.2, if u is an inward minimizer, we have on the other hand, using uχ R n \Br (x) as a competitor for u, we have Let now x ∈ K u and consider µ(r) = L n B r (x) ∩ { u > 0 } (1) .Using the isoperimetric inequality and inequality (3.5), we have that for almost every r ∈ (0, d(x, Ω)) Notice that we used Remark 3.3 in the last inequality.We have Integrating the differential inequality, we obtain which leads to a contradiction if δ 0 is too small, hence there exists a positive value δ 0 (Ω, β, p, q) such that { u > 0 } ⊂ B ρ(δ 0 ) .

Existence
In this section, we are going to prove the existence of a solution u to the problem (1.4).Let us denote We also denote by H 0 the set Proof.As we observed before, if u is a minimizer over H 0 then u is in H δ 0 , hence it is a minimizer over H δ 0 .Conversely, let us take u ∈ H δ 0 a minimizer over H δ 0 , and let us consider in addition v ∈ H 0 .Without loss of generality assume F(v) ≤ 2 F .We will prove that there exists a sequence w k of inward minimizers such that We first construct a family of functions v a ∈ H a such that with lim a→0 r(a) = 0. Let 0 < a < 1, and let v a = vχ { v≥a }∩B ρ(a) , where ρ(a) = a 1−q , we have In order to estimate the right-hand side, fix t ∈ (0, 1), and observe that by the coarea formula while, with a change of variables, By mean value theorem, for every k ∈ N we can find a k ≤ 1/k such that and in (4.1) we get We now construct the aforementioned sequence of inward minimizers: let us consider the functional with A containing Ω and contained in { v a k > 0 }.If we consider A j a minimizing sequence for G k , then they are certainly equibounded.Moreover, Notice in addition that since v a k ≥ a k on its support, then the jump set J χ A j va k clearly contains ∂ * A j .We now have that χ A j satisfies the conditions of Theorem 2.10, and eventually extracting a subsequence we can suppose that with a suitable A (k) , and moreover, letting w k = χ A (k) v a k , we have By construction w k is an inward minimizer, therefore we have w k ∈ H δ 0 , and consequently, we can compare it with u, obtaining Letting k go to infinity we get the thesis.
Proof.By Proposition 4.1 and Theorem 3.2 it is enough to find a minimizer in H δ 0 .Let u k be a minimizing sequence in H δ 0 , then, for k large enough, we have From Theorem 2.10 we have that there exists u ∈ H δ 0 such that, up to a subsequence, u k converges to u in L 1 loc and therefore u is a solution.
Proof of Theorem 1.1.The result is obtained by joining Proposition 4.2 and Theorem 3.2.

Density estimates
In this section, we prove the density estimates in Theorem 1.2 by adapting techniques used in [5] analogous to classical ones used in [7] to prove density estimates for the jump set of almost-quasi minimizers of the Mumford-Shah functional.
Definition 5.1.Let u ∈ SBV(R n ) be a function such that u = 1 a.e. in Ω.We say that u is a local minimizer for F on a set of finite perimeter Let E be a set of finite perimeter.We introduce the notation To prove Theorem 1.2 we will use the following Poincaré-Wirtinger type inequality whose proof can be found in [7, Theorem 3.1 and Remark 3.3].Let γ n be the isoperimetric constant relative to the balls of R n , i.e.
for every Borel set E, then Proposition 5.2.Let p ≥ 1 and let u ∈ SBV(B r ) such that Then there exist numbers −∞ < τ − ≤ m ≤ τ + < +∞ such that the function where the constants depend only on n, p, and r.For sufficiently small values of τ , there exist values r 0 , ε 0 depending only on n, τ, β, p, q and s such that, if r < r 0 , Proof.Without loss of generality, assume x = 0. Assume by contradiction that the conclusion fails, then for every τ > 0 there exists a sequence u k ∈ H s of local minimizers on B r k , with lim k r k = 0, such that and yet For every t ∈ [0, 1], we define the sequence of monotone functions By compactness of BV([0, 1]) in L 1 ([0, 1]), we can assume that, up to a subsequence, α k converges L 1 -almost everywhere to a monotone function α.Moreover, notice that, by (5.3), for every k α k (τ ) > τ n− 1 2 . (5.4) Our final aim is to prove that there exists a p-harmonic function v ∈ W 1,p (B 1 ) such that for every t Then v k ∈ SBV(B 1 ), and Thus, applying the Poincaré-Wirtinger type inequality in Proposition 5.2 to functions v k we obtain truncated functions ṽk and values m k , such that We prove that there exists v ∈ W 1,p (B 1 ) such that Step 1: and Notice that since ṽk is a truncation of v. From compactness theorems in SBV (see for instance [7, Theorem 3.5]), we have that ṽk − m k converges in L p (B 1 ) and L n -almost everywhere to a function v ∈ W 1,p (B 1 ), since H n−1 (J ṽk ) goes to 0 as k → +∞.Moreover, for every ρ < 1, since by definition Finally, up to a subsequence, Indeed, by (5.5), , which tends to zero as long as r p−1 k /E k is bounded.On the other hand, if r p−1 k /E k diverges, we could use the fact that ε k ≤ s −q F(u k ; B r k )r 1−n k and get which goes to zero.Using Fubini's theorem we have (5.7).
Let ũk (x) = E 1/p k ṽk ( x r k ), and for every t ∈ [0, 1] we define The αk functions are also monotone and bounded: the jump set of ũk is contained in J u k , therefore we can write using the fact that u k ∈ H s .As done for α k , we can assume that αk converges L 1 -almost everywhere to a function α.
Let I ⊂ [0, 1] be the set of values ρ for which (5.7) holds, α k and αk converge and α and α Step 2: are continuous.Notice that L 1 (I) = 1.Fix ρ, ρ ′ ∈ I with ρ < ρ ′ < 1 and let with ξ ∈ SBV(B 1 ).Let w ∈ W 1,p (B 1 ) and consider η a smooth cutoff function supported on B ρ ′ and identically equal to 1 in B ρ .Let We want to prove that and where R k goes to zero as k goes to infinity.We immediately compute and then we have (5.8).Now let and observe that ψ k coincides with u k outside B ρ ′ r k .We get from the local minimality of u k that (5.10) So, in particular, we have Dividing by F(u k ; B r k ) and using (5.7) we get With appropriate rescalings we have From (5.2) and the definition of E k , we have and then we get (5.9).
We want to prove that for every ϕ ∈ W 1,p (B 1 ) such that v − ϕ is supported on B ρ , we have Step 3: where C does not depend on either ρ or ρ ′ .From the definition of ϕ k , we have that on B ρ (5.12) We split the proof into two cases: either then v belongs to L ∞ (B 1 ) and there exists a positive constant C independent of k, and a natural number k such that for all k > k.Let ϕ ∈ W 1,p (B 1 ) with v − ϕ supported on B ρ , and let w = ϕ in the definition of ϕ k , then, for every k > k, we have L n -a.e. on B ρ ′ \ B ρ .From (5.15) we have that (5.16) Notice, in addition, that (5.12) reads as (5.17) finally joining (5.9), (5.17), (5.16), and (5.8), we have Letting k go to infinity we get (5.11).
We are now in a position to prove that v is p-harmonic: taking the limit as ρ ′ tends to ρ in (5.11), we have that if ϕ ∈ W for every ρ ∈ I, so that α is continuous on the whole interval [0, 1], α(1) = 1 and α(τ ) = lim k α k (τ ) ≥ τ n−1/2 .Nevertheless, if τ is sufficiently small this contradicts the fact that v is locally Lipschitz, since where C is a positive constant depending only on n and p.
We now prove that there exists a positive constant c = c(Ω, β, p, q) such that H n−1 (J u ∩ B r (x)) ≥ c(Ω, β, p, q)r n−1 (5.23) for every x ∈ K u and B r (x) ⊂ R n \ Ω. Assume by contradiction that there exists x ∈ J u such that, for r > 0 small enough, where ε 0 is the one in Lemma 5.3.Iterating Lemma 5.3 it can be proven (see [5,Theorem 5.1]) that lim with the Dirichlet condition u * = (βδ q ) −1/p on Ω.If Ω is sufficiently smooth we can apply the results in [4] to have that the density estimate for the jump set of minimizers holds up to the boundary of Ω.

Lemma 5 . 3 .
Let u ∈ H s be a local minimizer on B r (x) in the sense of definition Definition 5.1.