Abstract
In this work we address the question of the existence of nonradial domains inside a nonconvex cone for which a mixed boundary overdetermined problem admits a solution. Our approach is variational, and consists in proving the existence of nonradial minimizers, under a volume constraint, of the associated torsional energy functional. In particular we give a condition on the domain D on the sphere spanning the cone which ensures that the spherical sector is not a minimizer. Similar results are obtained for the relative isoperimetric problem in nonconvex cones.
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1 Introduction
In this paper we study an overdetermined problem for domains in a cone. This topic shares similarities with the question of characterising constant mean curvature hypersurfaces inside a cone (see [22, 23]) and hence with the isoperimetric problem. Thus we will also show some results for it.
Let D be a smooth domain on the unit sphere \({\mathbb {S}}^{N-1}\) and let \(\Sigma _D\) be the cone spanned by D, namely
For a domain \(\Omega \subset \Sigma _D\) we set
and assume that \({\mathcal {H}}_{N-1}(\Gamma _{1,\Omega })>0\), where \({\mathcal {H}}_{N-1}(\cdot )\) denotes the \((N-1)\)-dimensional Hausdorff measure. The set \(\Gamma _\Omega \) is usually called the relative (to \(\Sigma _D\)) boundary of \(\Omega \).
We consider the overdetermined mixed boundary value problem
for a constant \(c>0\), where \(\nu \) is the exterior unit normal. If \(\Gamma _\Omega \) is not smooth then the constant normal derivative condition is understood to hold on the regular part of \(\Gamma _\Omega \).
The overdetermined problem (1.2) arises naturally in the study of critical points of a relative torsional energy of subdomains of the cone \(\Sigma _D\) subject to a fixed volume contraint. Indeed, for any domain \(\Omega \), as in (1.2), let us consider the torsion problem with mixed boundary conditions
It is easy to see that (1.3) has a unique weak solution \(u_\Omega \) in the Sobolev space \(H_0^1(\Omega ; \Sigma _D)\) (see Sect. 6 or [9]), which is obtained by minimizing the functional
We then define the value
and we call it the torsional energy of \(\Omega \) in \(\Sigma _D\). Note that the second and third equality in (1.5) hold since \(u_\Omega \) is a weak solution of (1.3). By definition, the domain-dependent functional \(\Omega \mapsto {\mathcal {E}}(\Omega ; \Sigma _D)\) represents a relative version of the classical torsional energy functional usually defined using the solution of the analogous Dirichlet problem.
Using domain derivative techniques, as for other similar problems in shape-optimization theory it can be proved that the critical points of the functional \({\mathcal {E}}(\Omega ; \Sigma _D)\) with respect to volume-preserving deformations which leave the cone invariant, correspond to domains \(\Omega \) for which \(\frac{\partial u_\Omega }{\partial \nu }\) is constant on \(\Gamma _\Omega \), i.e. \(u_\Omega \) satisfies the overdetermined problem (1.2) (see [23, Proposition 4.3] if \(\Gamma _\Omega \) is smooth and \(u_\Omega \) has some Sobolev regularity, or Proposition 7.4 in the present paper in the nonsmooth case).
In this paper we intend to study the existence and the properties of domains for which a solution of (1.2) exists. It is easy to see that for any spherical sector \(\Omega _{D,R}:=B_R\cap \Sigma _D\), where \(B_R=B_R(0)\) is the ball with radius \(R>0\) centered at the origin (which is the vertex of the cone), the radial function
is a solution of (1.2) for \(\Omega =\Omega _{D,R}\). Therefore the question is whether the spherical sectors \(\Omega _{D,R}\) are the only domains for which (1.2) admits a solution. In the case of convex cones the answer was provided in [22], obtaining the following result (see [22, Theorem 1.1]):
Theorem 1.1
If \(\Sigma _D\) is convex, \(\Gamma _\Omega \) is smooth and u is a classical solution of (1.2) such that \(u \in W^{1,\infty }(\Omega )\cap W^{2,2}(\Omega )\), then
where \(B_R(P_0)\) is the ball centered at \(P_0\) with radius \(R=Nc\), and either \(P_0=0\), i.e. \(\Omega =\Omega _{D,R}\), or \(P_0\in \partial \Sigma _D\) and \(\Omega \) is a half-ball lying on a flat part of \(\partial \Sigma _D\).
Hence, if \(\Sigma _D\) is a convex cone, not flat anywhere, then the radial domains \(\Omega _{D,R}\) are the only domains admitting solutions of (1.2). Let us observe that the assumption \(u \in W^{1,\infty }(\Omega )\cap W^{2,2}(\Omega )\) can be seen as a “gluing condition". Indeed it is automatically satisfied whenever \(\Gamma _\Omega \) and \(\partial \Sigma _D\) intersect orthogonally (see [22, Sect. 6]).
In the context of the variational formulation of problem (1.2) described above, the result of Theorem 1.1 gives a characterization of the smooth critical points of \({\mathcal {E}}(\Omega ; \Sigma _D)\), restricted to the class of subdomains of fixed volume, in the case of convex cones. In particular any local minimizer of \({\mathcal {E}}(\Omega ; \Sigma _D)\) with a volume constraint is a spherical sector. Actually, using symmetrization methods in cones [19, 24] it can be proved (see [23]) that this holds in a more general class of cones which are the ones having an isoperimetric property.
In contrast, the case of nonconvex cones is largely unexplored, which is the main motivation of the present paper. The variational formulation of the overdetermined problem suggests that to look for nonradial domains for which there exists a solution of (1.2) is equivalent to look for nonradial critical points of \({\mathcal {E}}(\Omega ; \Sigma _D)\) under a volume constraint. In particular, if there are cones for which a minimizer of \({\mathcal {E}}(\Omega ; \Sigma _D)\) (fixing the volume) exists and if we are able to show that it is not the spherical sector then we achieve our goal. This is the content of our first main result.
Let us denote by \(\uplambda _1(D)\) the first nontrivial eigenvalue of the Laplace-Beltrami operator \(-\Delta _{{\mathbb {S}}^{N-1}}\) on D with zero Neumann boundary condition.
Theorem 1.2
If D is a smooth domain of \({\mathbb {S}}^{N-1}\) such that
where \({\mathbb {S}}^{N-1}_+\) is a half unit sphere, then there exists a bounded domain \(\Omega ^*\) which is a minimizer for \({\mathcal {E}}(\Omega ; \Sigma _D)\) with a fixed volume, but \(\Omega ^*\) is not a spherical sector \(\Omega _{D,R}\), for \(R>0\).
Moreover there exists a critical dimension \(d^*\) which can be either 5, 6 or 7, such that for the relative boundary \(\Gamma _{\Omega ^*}\) it holds that
-
(i)
\(\Gamma _{\Omega ^*}\) is smooth if \(N<d^*\);
-
(ii)
\(\Gamma _{\Omega ^*}\) can have countable isolated singularities if \(N=d^*\);
-
(iii)
\(\Gamma _{\Omega ^*}\) can have a singular set of dimension \(N-d^*\), if \(N>d^*\).
In addition on the regular part of \(\Gamma _{\Omega ^*}\) the normal derivative \(\frac{\partial u_{\Omega ^*}}{\partial \nu }\) is constant, where \(u_{\Omega ^*}\) is the torsion function of \(\Omega ^*\).
The condition \(\uplambda _1(D)<N-1\) in (1.7) is the one which ensures that a spherical sector \(\Omega _{D,R}\) cannot be a local minimizer for \({\mathcal {E}}(\Omega ; \Sigma _D)\) among the class of smooth subdomains of \(\Sigma _D\) with fixed volume, because it implies that it is not a stable critical point with respect to volume-preserving deformations (see Theorem 5.1). To prove this, we restrict the torsional energy functional to the class of strictly star-shaped sets \(\Omega \) in \(\Sigma _D\) with fixed volume \(c>0\), and we show the instability of the spherical sector \(\Omega _{D,R}\) with \(|\Omega _{D,R}|=c\) within this class. The reason to consider strictly star-shaped domains is that the relative boundary \(\Gamma _\Omega \) of a strictly star-shaped set is a radial graph of a function \(\varphi \) on D. This allows to study \({\mathcal {E}}(\Omega ; \Sigma _D)\) as a functional on \(\varphi \in C^2({\overline{D}})\).
On the other hand, the condition \({\mathcal {H}}_{N-1}(D)<{\mathcal {H}}_{N-1}({\mathbb {S}}^{N-1}_+)\) is the one which allows to prove the existence of a minimizer for \({\mathcal {E}}(\Omega ; \Sigma _D)\) (see Theorem 6.8 and Corollary 6.9). In the Appendix we give examples of domains D on \({\mathbb {S}}^{N-1}\) satisfying both conditions in (1.7).
Let us observe that, since \(\Sigma _D\) is not bounded, the existence of a minimizer for \({\mathcal {E}}(\Omega ; \Sigma _D)\) is not obvious. To prove Theorem 1.2 we use the concentration-compactness principle of P. L. Lions (see [17]). It was first used in shape-optimization Dirichlet problems in [6]. Having mixed boundary conditions, we cannot make use of the same proof as in [6]. We also stress that, as the cone \(\Sigma _D\) is not convex and since we do not have any information on the contact angle between \(\Sigma _D\) and \(\Gamma _{\Omega ^*}\), some care is needed to prove that the normal derivative \(\frac{\partial u_{\Omega ^*}}{\partial \nu }\) of the torsion function \(u_{\Omega ^*}\) is constant on the regular part of \(\Gamma _{\Omega ^*}\) (see Proposition 7.4). Finally, the regularity statements follow from the results of [11, 15] and [27].
As announced we also consider the isoperimetric problem in the cone to get a analogous nonradiality result using the same strategy.
The isoperimetric problem in the cone consists in minimizing the relative perimeter \({\mathcal {P}}(E; \Sigma _D)\) among all possibile finite relative perimeter sets E contained in the cone \(\Sigma _D\), with a fixed volume. It was proved in [18], and later in [10, 13, 25], that if \(\Sigma _D\) is a convex cone then the only minimizer of \({\mathcal {P}}(E; \Sigma _D)\) with a fixed volume are the spherical sectors \(\Omega _{D,R}\). This holds also in “almost" convex cones as shown in [2] (see also [23]). If the cone is not convex, a counterexample is given in [18].
Here we show that under the same conditions (1.7), a minimizer of \({\mathcal {P}}(E;\Sigma _D)\), exists, but is not the spherical sector \(\Omega _{D,R}\). Thus we have
Theorem 1.3
If D is a smooth domain of \({\mathbb {S}}^{N-1}\) such that (1.7) holds then there exists a bounded set of finite perimeter \(E^*\) inside \(\Sigma _D\) which minimizes \({\mathcal {P}}(E;\Sigma _D)\) for any fixed volume and \(E^*\) is not a spherical sector \(\Omega _{D,R}\), \(R>0\). Moreover for the relative boundary \(\Gamma _{E^*}\) it holds that
-
(i)
\(\Gamma _{E^*}\) can have a closed singular set \({\widetilde{\Gamma }}_{E^*}\) of Hausdorff dimension less than or equal to \(N-7\);
-
(ii)
\(\Gamma _{E^*}\setminus {\widetilde{\Gamma }}_{E^*}\) is a smooth embedded hypersurface with constant mean curvature;
-
(iii)
if \(x\in \overline{\Gamma _{E^*}\setminus {\widetilde{\Gamma }}_{E^*}} \cap \partial \Sigma _D\) then \(\Gamma _{E^*}\setminus {\widetilde{\Gamma }}_{E^*}\) is a smooth CMC embedded hypersurface with boundary in a neighborhood of x and meets \(\partial \Sigma _D\) orthogonally.
As for Theorem 1.2, the condition \(\uplambda _1(D)<N-1\) is the one which ensures that \(\Omega _{D,R}\) cannot be a local minimizer (see Theorem 8.3) and to prove this we again work in the class of smooth star-shaped sets. Instead the existence follows by results obtained in [25], while the regularity of minimizers derives from classical results for isoperimetric problems.
As a consequence of Theorem 1.3 we get that whenever (1.7) holds there exists a CMC hypersurface in the cone, namely \(\Gamma _{E^*}\), intersecting \(\partial \Sigma _D\) orthogonally, which is not a spherical cap centered at the vertex of the cone. It is important to notice that \(\Gamma _{E^*}\) cannot be a smooth radial graph. Indeed, by [22, Theorem 1.3] and [23, Theorem 1.1], we know that if \(\Gamma _{E^*}\) was a CMC radial graph intersecting \(\partial \Sigma _D\) orthogonally then \(E^*\) would be a spherical sector \(\Omega _{D,R}\), and this holds in any cone without requiring convexity hypotheses. It would be very interesting to understand what kind of CMC hypersurface \(\Gamma _{E^*}\) could be.
Finally we observe that, from our results and [18, Theorem 1.1] (or [22, Theorem 1.1]), we easily recover the inequality \(\uplambda _1(D)\geqq N-1\) whenever D is convex. This was proved in [12, Theorem 4.3] (see also [1, Theorem 4.1]).
The paper is organized as follows: in Sect. 2 we provide some geometric preliminaries. In Sect. 3 we study the torsional energy functional \({\mathcal {E}}(\Omega ; \Sigma _D)\) on strictly star-shaped domains in the cone, while in Sect. 4 we derive the formulas for the first and second variations of \({\mathcal {E}}(\Omega ; \Sigma _D)\) when the volume is fixed. In Sect. 5 we prove that the first condition in (1.7) allows to prove that the spherical sector is not a local minimizer for \({\mathcal {E}}(\Omega ; \Sigma _D)\). The long Sect. 6 is devoted to study the question of the existence of minimizers of \({\mathcal {E}}(\Omega ; \Sigma _D)\) with a volume constraint. Their properties are described in Sect. 7 where the proof of Theorem 1.2 is deduced. Finally in Sect. 8 we study the isoperimetric problem and prove Theorem 1.3. In the Appendix we give examples of nonconvex domains satisying the condition (1.7).
2 Some Preliminaries
In this section we fix some notation and we collect, for the reader’s convenience, some definitions and known facts from Riemannian Geometry that will be used throughout the paper.
Given a smooth manifold M, we denote by \(T_pM\) the tangent space at \(p\in M\), by \({\mathcal {T}}(M)\) the space of tangent vector fields on M and by TM the tangent bundle.
We denote by \(\langle .,\rangle \) or \(\varvec{\cdot }\) the standard scalar product in \({\mathbb {R}}^{N}\), by \(|\cdot |\) the Euclidean norm, and by \(\nabla ^0\) the flat connection of \({\mathbb {R}}^{N}\). In the special case \(M=D\), where \(D\subset {\mathbb {S}}^{N-1}\) is a domain of the unit sphere in \({\mathbb {R}}^N\), we denote by \(\nabla \) the induced Levi-Civita connection on D,namely
where \(\top :T{\mathbb {R}}^{N}\rightarrow TD\) is the orthogonal projection. If we further assume that D is a proper and smooth domain of \({\mathbb {S}}^{N-1}\) it will be always understood that D is considered as a submanifold with boundary, equipped with the induced Riemannian metric.
If \(\varphi :D\rightarrow {\mathbb {R}}\) is a smooth function, we adopt, respectively, the notations \(\mathrm{d}\varphi \), \(\nabla \varphi \), to indicate the differential and the gradient of \(\varphi \), which is the only vector field on D such that
We will also use sometimes the notation \(\nabla _{{\mathbb {S}}^{N-1}} \varphi \) instead of \(\nabla \varphi \) to make a distinction with respect to the usual gradient of real valued functions defined in open subsets of \({\mathbb {R}}^N\). The second covariant derivative of \(\varphi \) is defined as
and the Hessian of \(\varphi \), denoted by \(\nabla ^2 \varphi \) or by \(D^2 \varphi \), is the symmetric 2-tensor given by
The Laplacian of \(\varphi \), denoted by \(\Delta \varphi \), is the trace of the Hessian. Again, when there is a chance of confusion with the standard Laplacian we will use the notation \(\Delta _{{\mathbb {S}}^{N-1}} \varphi \) instead of \(\Delta \varphi \).
Let \(\{e_1,\ldots ,e_{N-1}\}\) be a local orthonormal frame field for D. For any \(i,j\in \{1,\ldots ,N-1\}\) we define the connection form \(\omega _{ij}\) as
We recall that the connection forms are skew symmetric and in terms of the \(\omega _{ij}\)’s we can write
We denote by \(\varphi _i\) the covariant derivative \(\nabla _{e_i}\varphi \), and we recall that, by definition, \(\nabla _{e_i} \varphi =\mathrm{d}\varphi [e_i]\). It is easy to check that the gradient of \(\varphi \) can be written as
Finally, taking \(X=e_i\), \(Y=e_j\) in (2.1) and using (2.2) we have
From now on we will use the notation \(\varphi _{ij}\) to denote \(\nabla _{e_i,e_j} \varphi \). In particular the Laplacian of \(\varphi \) can be written as \( \Delta \varphi = \sum _{i=1}^{N-1} \varphi _{ii}\).
Now we consider the special case of radial graphs.
Definition 2.1
Let \(D \subset {\mathbb {S}}^{N-1}\) be a domain and let \(\varphi \in C^2(D)\). We denote by \(\Gamma _\varphi \) the associated radial graph to \(\varphi \), namely
Clearly \(\Gamma _\varphi \) is a \((N-1)\)-dimensional manifold (of class \(C^2\)). We consider the map \({\mathcal {Y}}:D \rightarrow \Gamma _\varphi \) defined by
For any fixed \(q\in D\), let \(\gamma _i:(-\delta ,\delta ) \rightarrow D\), \(\gamma _i=\gamma _i(t)\) be a curve contained in D and such that \(\gamma _i(0)=q\), \(\gamma ^\prime _i(0)=e_i(q)\), for \(i=1,\ldots ,N-1\). Since
then a local basis for \(T_{{\mathcal {Y}}(q)}\Gamma _\varphi \) is given by
and the components of the induced metric are
We denote by \(\nu ({\mathcal {Y}}(q))\) the exterior unit normal at \({\mathcal {Y}}(q)\in \Gamma _\varphi \). It is easy to check that
In addition by direct computation we see that the coefficients of the second fundamental form are
for any \(i,j=1,\ldots ,N-1\) (see [20] or [4] for more details).
Finally, since the mean curvature at \({\mathcal {Y}}(q) \in \Gamma _\varphi \) is given by
where \((g^{ij})\) is the inverse matrix of \((g_{ij})\), namely
then, by a straightforward computation we see that \(\varphi \) must satisfy the following equation
Writing (2.9) in divergence form we obtain the well known equation for radial graphs of prescribed mean curvature (see [20] or [26])
3 Torsional Energy for Domains in Cones
In this section we define and study the torsional energy for smooth domains in cones and then we focus on the class of strictly star-shaped domains.
Let D be a smooth proper domain of \({\mathbb {S}}^{N-1}\) and let \(\Sigma _D\) be the cone spanned by D. For a bounded domain \(\Omega \subset \Sigma _D\) we set:
and assume that \({\mathcal {H}}_{N-1}(\Gamma _{1,\Omega })>0\) and that \(\Gamma _\Omega \) is a smooth \((N-1)\)-dimensional manifold whose boundary \(\partial \Gamma _\Omega =\partial \Gamma _{1,\Omega }\subset \partial \Sigma _D\setminus \{0\}\) is a smooth \((N-2)\)-dimensional manifold. The set \(\Gamma _\Omega \) is often called the relative (to \(\Sigma _D\)) boundary of \(\Omega \).
We consider the following mixed boundary value problem:
It is easy to see that (3.1) admits a unique weak solution \(u_\Omega \) in the space \(H_0^{1}(\Omega \cup \Gamma _{1,\Omega })\) which is the Sobolev space of functions in \(H^{1}(\Omega )\) whose trace vanishes on \(\Gamma _\Omega \). Indeed \(u_\Omega \) is the only minimizer of the functional
in the space \(H_0^{1}(\Omega \cup \Gamma _{1,\Omega })\) and we remark that \(u_\Omega >0\) a.e. in \(\Omega \), by the maximum principle (we refer to [22, 23] for more details).
Usually, the function \(u_\Omega \) is called torsion function of \(\Omega \) and its energy \(J(u_\Omega )\) represents the torsional energy of the domain \(\Omega \). This allows to consider the functional
which is defined on the domains contained in \(\Sigma _D\).
From the weak formulation of (3.1) we have
which implies that
Now we focus on the special case when \(\Omega \) is strictly star-shaped with respect to the origin which is the vertex of the cone \(\Sigma _D\). Thus we consider the relative boundary \(\Gamma _\Omega \) as the radial graph in \(\Sigma _D\) of a function \(\varphi \in C^2({{\overline{D}}}, {\mathbb {R}})\) as defined in Sect. 2. Therefore we denote \(\Omega \) by \(\Omega _\varphi \) which can be described as:
We restrict the torsional energy functional \({\mathcal {E}}\) to this class of domains and we denote it by \({\mathscr {E}}\), i.e. we set
We observe that \({\mathscr {E}}\) is a functional defined on \(C^2({\overline{D}},{\mathbb {R}})\) and we compute its first and second derivatives. To this aim we point out that taking variations of \(\varphi \) in \(C^2({\overline{D}},{\mathbb {R}})\) corresponds to taking variations of \(\Omega _\varphi \) in the class of strictly star-shaped domains (of class \(C^2\)).
Let us set for simplicity
If \(v \in C^2({{\overline{D}}}, {\mathbb {R}})\) and \(t \in (-\delta , \delta )\), where \(\delta >0\) is a fixed number, we consider the domain variations \(\Omega _{\varphi +tv} \subset \Sigma _D\), \({t\in (-\delta ,\delta )}\). Let \(\xi :(-\delta ,\delta )\times \Sigma _D \rightarrow \Sigma _D\) be the map defined by
It is elementary to check that, for a fixed \(t\in (-\delta ,\delta )\) the restriction
is a diffeomorphism whose inverse \(\left( \xi |_{\Omega _\varphi }\right) ^{-1}:\Omega _{\varphi +tv} \rightarrow \Omega _\varphi \) is given by
Moreover by definition we have \(\xi (t,x) \in \partial \Sigma _D\setminus \{0\}\) for all \((t,x) \in (-\delta ,\delta ) \times \partial \Sigma _D\setminus \{0\}\). In particular \(\xi \) is the flow associated to the vector field V on \(\Sigma _D\) given by
since \(\xi (0,x)=x\) and \(\frac{\mathrm{d}\xi }{\mathrm{d}t}(t,x)=e^{tv\left( \frac{x}{|x|}\right) } v\left( \frac{x}{|x|}\right) x=V(\xi (t,x))\), and \((\Omega _{\varphi +tv})_{t\in (-\delta ,\delta )}\) is a deformation of \(\Omega _\varphi \) associated to the vector field V (see [16, Definition 1.1]). We now compute the derivative of \({\mathscr {E}}\) with respect to a variation \(v \in C^2({{\overline{D}}},{\mathbb {R}})\).
Lemma 3.1
Let \(\varphi \in C^2({{\overline{D}}},{\mathbb {R}})\) and assume that \(u_{\Omega _\varphi }\in W^{1,\infty }(\Omega _\varphi )\cap W^{2,2}(\Omega _\varphi )\). Then, for any \(v \in C^2({{\overline{D}}},{\mathbb {R}})\), it holds that
where \(\mathrm{d}\sigma \) is the \((N-1)\)-dimensional area element of \({\mathbb {S}}^{N-1}\).
Proof
Let \(\varphi \in C^2({{\overline{D}}},{\mathbb {R}})\) as in the statement and let \(v \in C^2({{\overline{D}}},{\mathbb {R}})\). By definition we have
where \(u_{\Omega _{\varphi +tv}}\) is the only (positive) weak solution to
Writing (3.6) in polar coordinates we obtain that
Let \({\hat{\Phi }}:(-\delta ,\delta )\rightarrow H^{1}_0(\Omega _\varphi \cup \Gamma _{1,\varphi })\) be the map defined by
where \({\hat{u}}_t:=u_{\Omega _{\varphi +tv}}\circ \xi (t,\cdot )|_{\Omega _\varphi } \in H^{1}_0(\Omega _\varphi \cup \Gamma _{1,\varphi })\), \(\xi |_{\Omega _\varphi }(t,\cdot ):\Omega _\varphi \rightarrow \Omega _{\varphi +tv}\) is the diffeomorphism given by (3.4). From the proof of [23, Proposition 4.3] we know that \({\hat{\Phi }}\) is differentiable and thus we infer that \(u_{\Omega _{\varphi +tv}}\) is differentiable with respect to t. Hence, by the Leibniz integral rule for differentiation of integral functions we get that
In view of (3.7) we have \(u_{\Omega _{\varphi +tv}}(e^{\varphi +tv} q)=0\) on D for any \(t\in (-\delta , \delta )\). In particular computing at \(t=0\) we have
Setting \(u^\prime :=\left. \frac{\mathrm{d}}{\mathrm{d}t}\left( u_{\Omega _{\varphi +tv}}\right) \right| _{t=0}\) and arguing as in the proof of [23, Proposition 4.3], where the assumption \(u_{\Omega _\varphi }\in W^{1,\infty }(\Omega _\varphi )\cap W^{2,2}(\Omega _\varphi )\) is used, we infer that \(u^\prime \in H^{1}_0(\Omega _\varphi \cup \Gamma _{1,\varphi })\) satisfies
In particular, in view of (3.5) and since \(\Gamma _{\varphi }\) is a radial graph we have \(\nu (x)=\frac{\frac{x}{|x|}-\nabla _{{\mathbb {S}}^{N-1}}\varphi \left( \frac{x}{|x|}\right) }{\sqrt{1+|\nabla _{{\mathbb {S}}^{N-1}} \varphi \left( \frac{x}{|x|}\right) |^2}}\), for any \(x\in \Gamma _\varphi \) (see (2.7)), and thus
Rewriting (3.8) in terms of \(u^\prime \), applying Green’s second identity (which holds also in conic domains, since it is a consequence of the divergence theorem, see e.g. [22, Lemma 2.1]) and taking into account (3.1) (with \(\Omega =\Omega _\varphi \)), (3.9) and (3.10) we get that
where \(\mathrm{d}\sigma _{\Gamma _\varphi }\), \(\mathrm{d}\sigma _{\Gamma _{1,\varphi }\setminus \{0\}}\) are the \((N-1)\)-dimensional area elements of \(\Gamma _\varphi \), \(\Gamma _{1,\varphi }\setminus \{0\}\), respectively. Finally, writing \(x=e^{\varphi (q)}q\), \(q\in D\), observing that \(\mathrm{d}\sigma _{\Gamma _\varphi }=e^{(N-1)\varphi }\sqrt{1+|\nabla _{{\mathbb {S}}^{N-1}}\varphi |^2} \mathrm{d}\sigma \) and \(\frac{x}{|x|}=q\), then from (3.11) we obtain that
and this completes the proof. \(\square \)
For the second variation of the functional \({\mathscr {E}}\) we have
Lemma 3.2
Let \(\varphi \) be as in Lemma 3.1. Then, for any \(v,w \in C^2({{\overline{D}}}, {\mathbb {R}})\), it holds that
where \(u^\prime _w=\left. \frac{\mathrm{d}}{\mathrm{d}s}\left( u_{\Omega _{\varphi +sw}}\right) \right| _{s=0}\) is the solution to (3.9) with V given by \(V(x)=w\left( \frac{x}{|x|}\right) x\).
Proof
Let us fix \(v, w \in C^2({{\overline{D}}}, {\mathbb {R}})\), by definition and by Lemma 3.1 we have
and thus
Since \(\Gamma _{\varphi +sw}\) is a radial graph, then, in view of (2.7), we have
As in the proof of Lemma 3.1 we consider the map \({\hat{\Phi }}:(-\delta ,\delta )\rightarrow H^{1}_0(\Omega _\varphi \cup \Gamma _{1,\varphi })\), defined by
Moroever, let \(G:H^{1}_0(\Omega _\varphi \cup \Gamma _{1,\varphi }) \rightarrow L^2(\Omega _\varphi \cup \Gamma _{1,\varphi }, {\mathbb {R}}^N)\), given by \(G(f):=\nabla f\). Since G is a bounded linear operator, then G is differentiable, \(G^\prime (f)[g]=\nabla g\) for any \(g \in H^{1}_0(\Omega _\varphi \cup \Gamma _{1,\varphi })\). In addition, as \({\hat{\Phi }}\) is differentiable (see the proof of [23, Proposition 4.3] for the details), then the composition \(G\circ {\hat{\Phi }}: (-\delta ,\delta )\rightarrow L^2(\Omega _\varphi \cup \Gamma _{1,\varphi }, {\mathbb {R}}^N)\) is differentiable and
In terms of \({\hat{u}}_s\), this means that
In addition, since \(u_{\Omega _{\varphi +sw}}={\hat{u}}_s \circ \xi (-s,\cdot )|_{\Omega _{\varphi +sw}}\) it follows that also \(s\mapsto \nabla u_{\Omega _{\varphi +sw}}\) is differentiable. We claim that
Indeed, setting \(\xi _s:=\xi (-s,\cdot )|_{\Omega _{\varphi +sw}}\), since
then, using (3.16), we get that
and, thus by a straightforward computation, we obtain
for any \(i=1,\ldots ,N\), which proves Claim (3.17).
Thanks to (3.15) and (3.17) we have
Finally, combining (3.14) and (3.18) we readily obtain (3.12). The proof is complete.
\(\square \)
4 Volume-Constrained Critical Points for the Torsional Energy of Star-Shaped Domains
For any \(\varphi \in C^2({{\overline{D}}},{\mathbb {R}})\), the volume of the associated star-shaped domain \(\Omega _\varphi \) (see (3.3)) is given by
where \(\mathrm{d}\sigma \) is the \((N-1)\)-dimensional area element of \({\mathbb {S}}^{N-1}\). It is easy to check that \({\mathcal {V}}\) is of class \(C^2\) and for any \(v,w \in C^2({{\overline{D}}},{\mathbb {R}})\) it holds
and
For a number \(c>0\) we define
Clearly M is a smooth manifold and for any \(\varphi \in M\) it holds
We consider the restriction of the torsional energy to the domains corresponding to functions \(\varphi \in M\), namely the functional defined by
If \(\varphi \in M\) is critical point of I then there exists \(\uplambda \in {\mathbb {R}}\) such that
As a straightforward consequence of Lemma 3.1 and (4.2) we have
Lemma 4.1
Let \(\varphi \in M\) be a critical point for I and assume that \(u_{\Omega _\varphi }\in W^{1,\infty }(\Omega _\varphi )\cap W^{2,2}(\Omega _\varphi )\). Then the Lagrange multiplier \(\uplambda \) is negative and
Proof
Let \(\varphi \in M\) be a critical point for I and assume that \(u_{\Omega _\varphi }\in W^{1,\infty }(\Omega _\varphi )\cap W^{2,2}(\Omega _\varphi )\), then, from (4.7) and exploiting Lemma 3.1 and (4.2), we have
for any \(v \in C^2({{\overline{D}}}, {\mathbb {R}})\). Hence we readily obtain that
and from the arbitrariness of \(v \in C^2({{\overline{D}}}, {\mathbb {R}})\) we easily deduce that \(\uplambda <0\) and
Now, recalling that \(u_{\Omega _\varphi }\) is the only (positive) weak solution to
then from standard regularity estimates we infer that \(u_{\Omega _\varphi }\) is smooth in \(\Omega _\varphi \), and from Hopf’s lemma we get that \(\frac{\partial u_{\Omega _\varphi } }{\partial \nu }<0\) on \(\Gamma _\varphi \). Hence, in view of (4.8), we obtain
\(\square \)
Remark 4.2
From Lemma 4.1 we deduce that each critical point of I produces a star-shaped domain \(\Omega _\varphi \) for which the overdetermined problem (1.2) has a solution. We recall that, as shown in [23, Proposition 4.3], each critical point of the functional \({\mathcal {E}}(\Omega ; \Sigma _D)\) on the whole family of domains in \(\Sigma _D\), with a volume constraint, is a domain for which (1.2) has a solution. Hence Lemma 4.1 shows that the same statement holds even if the variations are taken only in the class of star-shaped domains.
In the next result we compute the second derivative of I at critical point along variations in \(T_\varphi M\).
Lemma 4.3
Let \(\varphi \in M\) be a critical point for I and let \(v,w \in T_\varphi M\). Then
where \(\uplambda \) is the Lagrange multiplier.
Proof
By definition if \(\varphi \in M\) is a critical point for I, the second variation \(I^{\prime \prime }(\varphi )[v,w]\) along the variations \(v,w \in T_\varphi M\) is given by
where \(\Psi :(-\epsilon ,\epsilon )\times (-\epsilon ,\epsilon )\rightarrow M\) is a smooth surface in M such that
We recall that by definition it holds \(I(\Psi (t,s))= {\mathscr {E}}(\Psi (t,s))\). Since
we have
On the other hand, since \(\Psi (t,s) \in M\) we have \({\mathcal {V}}(\Psi (t,s))=c\) for any \((t,s)\in (-\epsilon ,\epsilon )\times (-\epsilon ,\epsilon )\), and thus differentiating with respect to t we infer that \({\mathcal {V}}^\prime (\Psi (t,s))[\frac{\partial \Psi }{\partial s}(t,s)]=0\). Differentiating again with respect to s we obtain
Hence, computing (4.10) at \((t,s)=(0,0)\), since \(\varphi =\Psi (0,0)\) is a critical point of I and taking into account that (4.7), (4.11), we get that
which proves the desired relation. \(\square \)
Remark 4.4
When \(\varphi \equiv 0\) then \(\Omega _\varphi \) is the unit spherical sector \(\Omega _{D,1}=\Sigma _D\cap B_1\), where \(B_1=B_1(0)\) is the unit ball in \({\mathbb {R}}^N\) centered at the origin. We denote it by \(\Omega _0\), while \(\Gamma _0\) will be its relative boundary. In this case the torsion function \(u_{\Omega _0}\) is known to be the radial function \(u_{\Omega _0}(x)=\frac{1-|x|^2}{2N}\). Then we can choose \(c=|\Omega _0|\) in the definition of M and the tangent space to M at \(\varphi \equiv 0\) is \(T_0M=\{v \in C^2({{\overline{D}}}, {\mathbb {R}});\ \int _D v \ \mathrm{d}\sigma =0 \}\). It is easy to check that \(\nabla u_{\Omega _0}=-\frac{1}{N}x\), for \(x \in \Sigma _D\cap B_1\), and \(\frac{\partial u_{\Omega _0}}{\partial \nu }=-\frac{1}{N}\) on \(\Gamma _0\), so that \(\Omega _0\) is a critical point for I with \(\uplambda =-\frac{1}{2N^2}\). Finally \(D^2 u_{\Omega _0}(x)=-\frac{1}{N} {\mathbb {I}}_N\), for \(x \in \Sigma _D\cap B_1\), where \({\mathbb {I}}_N\) is the identity matrix of order N, and thus we readily have that \(u_{\Omega _0}\in W^{1,\infty }(\Omega _0)\cap W^{2,2}(\Omega _0)\).
For the second variation we have
Proposition 4.5
For any \(v \in T_0M\) it holds that
where \(u^\prime =\left. \frac{\mathrm{d}}{\mathrm{d}t}\left( u_{\Omega _{0+tv}}\right) \right| _{t=0}\) (see (3.9)) and \(\frac{\partial u^\prime }{\partial \nu }\) is the normal derivative of \(u^\prime \) on \(\Gamma _0=D\).
Proof
First we observe that, taking \(\varphi \equiv 0\), from (3.9) and (3.5) we have that \(u^\prime \) satisfies
Then, taking \(v \in C^2({{\overline{D}}}, {\mathbb {R}})\) such that \(\int _D v \ \mathrm{d}\sigma =0\), from Lemma 3.2, Lemma 4.3, Remark 4.4 and (4.3), (4.13) we obtain
since \(\nabla u_{\Omega _0} \varvec{\cdot } \nabla _{{\mathbb {S}}^{N-1}} v\equiv 0\) in D because \(\nabla u_{\Omega _0}\) is proportional to the radial direction. \(\square \)
Remark 4.6
We observe that thanks to (4.12), since \(u^\prime =\frac{1}{N} v\) on \(\Gamma _0\), by (4.13) and recalling that \(\Gamma _0=D\), we can write
Then by Green’s identity and (4.13) we infer that
5 A Condition for Instability
In this section we provide conditions on the domain \(D\subset {\mathbb {S}}^{N-1}\) such that the corresponding spherical sector (i.e. the domain \(\Omega _0\) associated to the function \(\varphi \equiv 0\), see (3.3)) is not a local minimizer for the torsional energy functional under a volume constraint. This is achieved by showing that \(\Omega _0\) is an unstable critical point of I, i.e. its Morse index is positive.
More precisely, let M be the manifold defined in (4.4), with \(c=|\Omega _0|\) and let I be as in (4.6). As observed in Remark 4.4 the function \(\varphi \equiv 0\) belongs to M and \(\Omega _0\) is a critical point for I. The main result of this section is the following:
Theorem 5.1
Let \(D\subset {\mathbb {S}}^{N-1}\) be a smooth domain and let \(\uplambda _1(D)\) be the first non trivial eigenvalue of the Laplace-Beltrami operator \(-\Delta _{{\mathbb {S}}^{N-1}}\), with zero Neumann condition on \(\partial D\). It holds that
-
(i)
if \(\uplambda _1(D) < N-1\), then \(\Omega _0\) is not a local minimizer for I;
-
(ii)
if \(\uplambda _1(D) > N-1\), then \(\Omega _0\) is a local minimizer for I.
Proof
To prove (i), let \((w_j)_{j\in {\mathbb {N}}}\) be a \(L^2(D)\)-orthonormal basis of eigenfunctions of the eigenvalue problem
where \(\nu _{_{\partial D}}\) is the exterior unit co-normal to \(\partial D\), i.e. for any \(q\in \partial D\), \(\nu _{_{\partial D}}(q)\) is the only unit vector in \(T_q{\mathbb {S}}^{N-1}\) such that \(\nu _{_{\partial D}}(q)\perp T_q\partial D\) and \(\nu _{_{\partial D}}(q)\) points outward D. We define the following extension of \(w_j\) to the cone \(\Sigma _D\)
where
We claim that \(w={\tilde{w}}_j\big |_{\Omega _0}\) is the unique solution of
Indeed, writing the Laplace operator in polar coordinates and exploiting (5.1) we easily check that
because \(\alpha _j\) satisfies \(\alpha _j^2 + (N-2)\alpha _j -\uplambda _j=0\). Moreover, by definition, we have \({\tilde{w}}_j\big |_D=\frac{1}{N}w_j\) and \(\frac{\partial {\tilde{w}}_j}{\partial \nu } = 0\) on \(\Gamma _{1,0}\).
Now, let us take \(j=1\). It is well known that the first eigenfunction \(w_1\) is smooth and satisfies \(\int _D w_1 \ \mathrm{d}\sigma =0\), i.e. \(w_1\in T_0M\). Computing \(I^{\prime \prime }(0)[w_1,w_1]\), thanks to Proposition 4.5 and taking into account that \({\tilde{w}}_1\big |_{\Omega _0}\) is the solution of (5.4), with \(j=1\), we get that
Then, since the \(L^2(D)\)-norm of \(w_1\) is equal to 1, the exterior unit normal \(\nu \) to \(\Gamma _0\) is the radial direction, and
from (5.5) we obtain
Thus we deduce that
Finally, from (5.3) it is immediate to check that \(\alpha _1<1\) is equivalent to \(\uplambda _1(D)<N-1\) and the proof of (i) is complete.
To prove (ii), let \(v\in T_0M\) and assume, without loss of generality, that \(\int _D v^2 \ \mathrm{d}\sigma =1\). Taking \((w_j)_{j\in {\mathbb {N}}}\) as in the proof of (i), since \(v \in T_0 M\) we can write
Let \({\tilde{w}}_j\) be the harmonic extension of \(w_j\) defined in (5.2). Then, as \({\tilde{w}}_j\big |_{\Omega _0}\) is a solution to (5.4) for any \(j\in {\mathbb {N}}\), we infer that \({\tilde{v}}:=\sum _{j=1}^\infty (v,w_j)_{L^2(D)} {\tilde{w}}_j\) is a solution to
Thus, by Proposition 4.5, we get
As in (5.6) we have that \(\frac{\partial {\tilde{w}}_j}{\partial \nu }=\frac{\alpha _j}{N} w_j\) on D, for any \(j\in {\mathbb {N}}\). Hence, since \(\int _D v^2 \ \mathrm{d}\sigma =1\), we deduce
Now, if \(\uplambda _1(D)>N-1\) it follows that \(\alpha _1>1\), and, as \((\alpha _j)_{j\in {\mathbb {N}}}\) is a nondecreasing sequence, we obtain
having used that \(\sum _{j=1}^\infty (v,w_j)_{L^2(D)}^2=1\), as \(\int _D v^2(q) \ \mathrm{d}\sigma =1\). Hence (ii) holds. \(\square \)
We conclude this section with a useful criterion for checking the property \(\uplambda _1(D)<N-1\). To this end let \({e} \in {\mathbb {S}}^{N-1}\) and let \(u_{{e}}\in C^\infty ({\mathbb {R}}^N)\) be the function defined by
which satisfies
We have
Proposition 5.2
Let D be a smooth proper domain of \({\mathbb {S}}^{N-1}\) and let \({e}\in {\mathbb {S}}^{N-1}\) satisfy
Assume that either one of the following holds:
-
(i)
\(\displaystyle \int _{\partial D} u_{{e}} \frac{\partial u_{{e}}}{\partial \nu } \ \mathrm{d}{\hat{\sigma }}<0\);
-
(ii)
\(\displaystyle \int _{\partial D} u_{{e}} \frac{\partial u_{{e}}}{\partial \nu } \ \mathrm{d}{\hat{\sigma }}=0\), and \(u_{{e}}\) is not an eigenfunction of \( {\left\{ \begin{array}{ll} -\Delta _{{\mathbb {S}}^{N-1}} w=\uplambda w&{}{} \text{ in }\ D,\\ \quad \quad \quad \quad \frac{\partial w}{\partial \nu } =0 &{}{} \text{ on }\ \partial {D},\end{array}\right. }\)
where \(\mathrm{d}{\hat{\sigma }}\) is the \((N-2)\)-dimensional area element of \(\partial D\) and \(\nu =\nu _{_{\partial D}}\) is the exterior unit co-normal to \(\partial D\). Then \(\uplambda _1(D)<N-1\).
Proof
Taking \(u_{{e}}\) as test function in the variational characterization of the first non-trivial eigenvalue of \(-\Delta _{{\mathbb {S}}^{N-1}}\) with zero Neumann condition on \(\partial D\), applying Green’s identity and exploiting (5.9), we have
Therefore, if (i) holds it follows that
which implies that \(\uplambda _1(D)<N-1\). This completes the proof for the case (i). On the other hand, under the assumption (ii), the equality sign in (5.10) holds, but as \(u_{{e}}\) is not an eigenfunction it follows that \(N-1\) cannot be the smallest non-trivial eigenvalue. \(\square \)
6 Existence of Volume-Constrained Minimizers for the Torsional Energy
Let \(D\subset {\mathbb {S}}^{N-1}\) be a domain of the unit sphere and let \(\Sigma _D\) be the cone generated by D. We will always assume that D is smooth so that \(\Sigma _D\) is smooth exept at the vertex. In Sect. 3 we defined the torsional energy \({\mathcal {E}}(\Omega ;\Sigma _D)\) for smooth domains \(\Omega \subset \Sigma _D\) strictly star-shaped with respect to the vertex of the cone. In this section we study the minimization problem for the torsional energy under a volume constraint in a larger class of sets. Thus we recall some definitions.
Definition 6.1
We say that \(\Omega \subset {\mathbb {R}}^N\) is quasi-open, if for any \(\varepsilon >0\), there exists an open set \(\Lambda _\varepsilon \) such that \(\mathrm {cap}(\Lambda _\varepsilon )\leqq \varepsilon \) and \(\Omega \cup \Lambda _\varepsilon \) is open, where \(\mathrm {cap}(\Lambda _\varepsilon )\) denotes the capacity of \(\Lambda _\varepsilon \) with respect to the \(H^1\)-norm (see [14, Sect. 3.3] or [9, Sect. 2.1]).
For any quasi-open set \(\Omega \subset \Sigma _D\) we consider the Sobolev space:
where q.e. means quasi-everywhere, i.e. up to sets of zero capacity.
Definition 6.2
We say that u is a (weak) solution of the mixed boundary value problem
if \(u\in H^1_0(\Omega ; \Sigma _D)\) and
Remark 6.3
As \(\Sigma _D\) is connected and smooth (execpt at the vertex) then \(\Sigma _D\) is uniformly Lipschitz. Thus if \(|\Omega |<+\infty \) the inclusion \(H^1_0(\Omega ; \Sigma _D)\hookrightarrow L^2(\Sigma _D)\) is compact (see [9, Proposition 2.3-(i)]). This implies that the functional
has a unique minimizer \(u_\Omega \in H^1_0(\Omega ; \Sigma _D)\) which is the unique (weak) solution to (6.1), which is called energy function or torsion function of \(\Omega \). We also recall that \(\Omega =\{u_\Omega >0\}\) up to a set of zero capacity (see [9, Proposition 2.8-(e)]). Moreover we denote by \(\uplambda _1(\Omega ; \Sigma _D)\) the first eigenvalue of the Laplacian in \(H_0^1(\Omega ; \Sigma _D)\), i.e.
Then, as before, we define the torsional energy of \(\Omega \) (relative to \(\Sigma _D\)) as:
We want to study the problem of minimizing the functional \({\mathcal {E}}(\Omega ;\Sigma _D)\) among quasi-open sets of uniformly bounded measure. Therefore, fixing \(c>0\) we define
Our aim is to give a sufficient condition on the cone \(\Sigma _D\) (hence on D) for the infimum in (6.5) to be achieved. We begin by recalling some known properties of the function \(u_\Omega \) that will be used in this section.
Proposition 6.4
Let \(c>0\). There exists a positive constant C depending only on N, \(\Sigma _D\) and c such that, for any quasi-open subset \(\Omega \) of \(\Sigma _D\) with \(|\Omega | \leqq c\), it holds that
-
(i)
\(u_\Omega \) is bounded and \(\Vert u_\Omega \Vert _{L^\infty (\Sigma _D)}\leqq C |\Omega |^{2/N}\);
-
(ii)
\(\int _{\Sigma _D} |\nabla u_\Omega |^2\ \mathrm{d}x \leqq C |\Omega |^{\frac{N+2}{N}}\);
-
(iii)
\(\int _{\Sigma _D} u_\Omega ^2\ \mathrm{d}x \leqq C |\Omega |^{\frac{N+4}{N}}\).
Proof
Let us fix \(c>0\). Since the cone \(\Sigma _D\) is a uniformly Lipschitz connected open set of \({\mathbb {R}}^N\) then we can apply [9, Lemma 2.5]. Hence, for any quasi-open subset \(\Omega \subset \Sigma _D\), with \(|\Omega |\leqq c\), fixing \(p \in ]N/2, +\infty [\) and taking \(f=\chi _\Omega \), where \(\chi _\Omega \) denotes the characteristic function of \(\Omega \), we obtain from [9, Lemma 2.5] that there exists a positive constant \({\tilde{C}}\) depending on N, p, \(\Sigma _D\) and c only such that
which gives (i).
Next, taking \(u_\Omega \) as test function in the weak formulation of (6.1) we get
and, by (i), we obtain
i.e. (ii). Finally (iii) is a trivial consequence of (i) since
\(\square \)
Notice that as a straightforward consequence of the previous result it holds that \(\mathcal {O}_c(\Sigma _D)>-\infty \).
Remark 6.5
As remarked in [23, Remark 4.2] there is a natural invariance by scaling in our problem, which, in particular, allows to claim that the infimum as in (6.5), but with volume bounded by another constant \(\uplambda >0\), can be easily computed from \(\mathcal {O}_c(\Sigma _D)\). Namely we have
Indeed, for any quasi-open \(\Omega \subset \Sigma _D\), for any \(t>0\) it holds that \(t\Omega \subset \Sigma _D\), \(|t\Omega |=t^N|\Omega |\), and it is easy to check that \(u_{t\Omega }(x)=t^2 u_\Omega \left( \frac{x}{t}\right) \) and
In particular \(\mathcal {O}_c(\Sigma _D)\) can be defined by taking \(|\Omega |=c\) in (6.5) and either a minimizer exists for any fixed volume or there are no minimizers whatever bound for the volume is chosen.
Among the quasi-open sets in \(\Sigma _D\) we can consider the spherical sectors
In this case the solution of (6.1) is radial and explicitly given by
and its energy is
Therefore, by (6.5), we have
where \(R_c=R_c(D)>0\) is such that \(|\Omega _{D,R_c}|=R_c^{N} {\mathcal {H}}_{N-1}(D)=c\), namely \(R_c(D)=\left( \frac{c}{ {\mathcal {H}}_{N-1}(D)}\right) ^{\frac{1}{N}}\).
Remark 6.6
Notice that for any \(c>0\) it holds
which means that \({\mathcal {E}}(\Omega _{D,R_c}; \Sigma _D)\) is monotone increasing with respect to \({\mathcal {H}}_{N-1}(D)\).
Remark 6.7
When D is a hemisphere, let us say for convenience the upper hemisphere, denoted by \({\mathbb {S}}^{N-1}_+={\mathbb {S}}^{N-1}\cap \{(x_1,\ldots ,x_N)\in {\mathbb {R}}^N; \ x_N>0\}\), then the cone \(\Sigma _{{\mathbb {S}}^{N-1}_+}\) coincides with the half-space \({\mathbb {R}}^N_+\). In this case it is well known, for example by symmetrization, that \(\mathcal {O}_c\left( \Sigma _{{\mathbb {S}}^{N-1}_+}\right) \) is achieved by any half-ball of measure c and
In the general case, using the smoothness of the cone, we prove in Proposition 6.10 that it always holds
The main result of this section is to show that if the strict inequality holds in (6.14) then the infimum is achieved. Indeed we have
Theorem 6.8
Let \(c>0\) and assume that
Then \(\mathcal {O}_c(\Sigma _D)\) is achieved.
Proof
Let \((\Omega _n)_n \subset \Sigma _D\) be a minimizing sequence for \(\mathcal {O}_c(\Sigma _D)\) and consider the corresponding energy functions \(u_{\Omega _n}\in H^1_0(\Omega _n;\Sigma _D)\) for any \(n\in {\mathbb {N}}\). By definition, we have
Setting \(u_n:=u_{\Omega _n}\), since \(|\Omega _n|\leqq c\), for any \(n\in {\mathbb {N}}\), by Proposition 6.4 we find a positive constant \(C_1\) independent of n such that
In particular, up to a subsequence (still denoted by \((u_n)_n\)), we have \(\Vert u_n\Vert _{L^2(\Sigma _D)}^2\rightarrow \uplambda \), for some \(\uplambda \geqq 0\). We first observe that \(\uplambda >0\). Otherwise, if \(\uplambda =0\), by Hölder’s inequality and exploiting the uniform bound \(|\Omega _n|\leqq c\), we would have
which implies that \({\mathcal {E}}(\Omega _n; \Sigma _D)\rightarrow 0\), as \(n\rightarrow +\infty \), contradicting \(\mathcal {O}_c(\Sigma _D)<0\) (see (6.11)). Therefore, as \((u_n)_n\) is bounded in \(H^1(\Sigma _D)\) and
for some \(\uplambda >0\), we can apply, with small modifications in the proof, the concentration-compactness principle of P. L. Lions (see [17, Lemma III.1]). Hence, there exists a subsequence \((u_{n_k})_k\) satisfying one of three following possibilities:
-
(i)
there exists \((y_{n_k})_k \subset \overline{\Sigma _D}\) satisfying
$$\begin{aligned} \forall \varepsilon>0 \ \exists R>0 \ \ \hbox {such that} \ \ \int _{B_R(y_{n_k})\cap \Sigma _D} u_{n_k}^2 \ \mathrm{d}x \geqq \uplambda - \varepsilon \ \ \forall k\in {\mathbb {N}}; \end{aligned}$$ -
(ii)
\(\displaystyle \lim _{k\rightarrow +\infty } \sup _{y \in \Sigma _D} \int _{B_R(y)\cap \Sigma _D} u_{n_k}^2 \ \mathrm{d}x=0\), for all \(R>0\);
-
(iii)
there exists \(\alpha \in ]0,\uplambda [\) such that for all \(\varepsilon >0\), there exist \(k_0\geqq 1\) and two sequences \((u_{1,k})_k\), \((u_{2,k})_k\) bounded in \(H^1(\Sigma _D)\) satisfying, for \(k\geqq k_0\)
$$\begin{aligned}&\Vert u_{n_k} - u_{1,k}-u_{2,k}\Vert _{L^2(\Sigma _D)} \leqq 4\varepsilon ,\\&\left| \int _{\Sigma _D} u_{1,k}^2 \ \mathrm {d}x -\alpha \right| \leqq \varepsilon , \ \ \left| \int _{\Sigma _D} u_{2,k}^2 \ \mathrm {d}x -(\uplambda -\alpha ) \right| \leqq \varepsilon ,\\&\text{ dist }(\text{ supp }(u_{1,k}), \text{ supp }(u_{2,k})) \rightarrow +\infty , \ \text{ as } k\rightarrow +\infty ,\\&\liminf _{k\rightarrow +\infty } \int _{\Sigma _D} |\nabla u_{n_k}|^2 - |\nabla u_{1,k}|^2 - |\nabla u_{2,k}|^2 \ \mathrm {d}x \geqq 0. \end{aligned}$$
We now divide the proof in some steps. We begin by showing that the “vanishing” case (ii) cannot occur.
Step 1: (ii) cannot happen.
Assume by contradiction that (ii) holds. The idea is to show that \(u_{n_k}\rightarrow 0\) strongly in \(L^2(\Sigma _D)\), as \(k\rightarrow +\infty \), contradicting (6.18). To prove this we invoke [28, Lemma 1.21] (whose proof can be easily adapted for functions in \(H^1(\Sigma _D)\)), which claims that (ii) and (6.16) imply \(u_{n_k}\rightarrow 0\) in \(L^p(\Sigma _D)\), for any \(2<p<2^*\), as \(k\rightarrow +\infty \), where \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent. Then, exploiting that \(u_{n_k}\in H_0^1(\Omega _{n_k}; \Sigma _D)\) and \(|\Omega _{n_k}|\leqq c\), by Hölder’s inequality we readily conclude that \(u_{n_k}\rightarrow 0\) in \(L^2(\Sigma _D)\), as \(k\rightarrow +\infty \).
In the next step we prove that the “dichotomy” case (iii) cannot occur.
Step 2: (iii) cannot happen.
Assume by contradiction that (iii) holds. We claim that, up to a further subsequence, there exists another minimizing sequence \(({{\widetilde{\Omega }}}_{n_k})_k\subset \Sigma _D\), with \({{\widetilde{\Omega }}}_{n_k} \subset \Omega _{n_k}\), for any k, satisfying:
-
\({{\widetilde{\Omega }}}_{n_k} = \Omega _{1,k} \cup \Omega _{2,k}\), for some quasi-open subsets \(\Omega _{1,k}\), \(\Omega _{2,k}\) of \(\Omega _{n_k}\);
-
\(\mathrm {dist}( \Omega _{1,k}, \Omega _{2,k})\rightarrow +\infty \), as \(k\rightarrow +\infty \);
-
\(c_i:=\liminf _{k\rightarrow +\infty } |\Omega _{i,k}|>0\), for \(i=1,2\).
Indeed, by (iii) and a diagonal argument, we find bounded subsequences \((u_{1,k})_k\), \((u_{2,k})_k\) in \(H^1(\Sigma _D)\) (still indexed by k) satisfying
By the proof of [17, Lemma III.1] we see that \(u_{1,k}\), \(u_{2,k}\) can be chosen to be non-negative and in addition, since \(u_{n_k} \in H^1_0(\Omega _{n_k};\Sigma _D)\), we also have that \(u_{1,k}, u_{2,k} \in H^1_0(\Omega _{n_k};\Sigma _D)\) for any k. In particular, as \(u_{1,k}, u_{2,k} \in H^1(\Sigma _D)\), setting \(\Omega _{1,k}:=\{u_{1,k}>0\}\), \(\Omega _{2,k}:=\{u_{2,k}>0\}\) it follows that \(\Omega _{1,k}\), \(\Omega _{2,k}\) are quasi-open subsets of \(\Sigma _D\). Therefore, \({{\widetilde{\Omega }}}_{k}:= \Omega _{1,k} \cup \Omega _{2,k}\) is a quasi-open set contained in \(\Omega _{n_k}\) and denoting by \({\tilde{u}}_{n_k}:=u_{{{\widetilde{\Omega }}}_{n_k}}\) the torsion function of \({{\widetilde{\Omega }}}_{n_k}\) and arguing as in [6, Sect. 3.3] (with obvious small modifications), we infer that
From (6.20) it follows that \(({{\widetilde{\Omega }}}_{n_k})_k\) is a minimizing sequence for \({\mathcal {O}}_c(D)\). Moreover, by construction and (6.19) we readily deduce that \(\mathrm {dist}( \Omega _{1,k}, \Omega _{2,k})\rightarrow +\infty \), as \(k\rightarrow +\infty \). Finally, setting
it holds that \(c_i>0\) for \(i=1,2\). Indeed, assuming by contradiction, for instance, that \(c_1=0\), by Hölder’s inequality and Sobolev’s inequality (note that \(\Sigma _D\) satisfies the cone condition) we get
Now, recalling that \((u_{1,k})_k\) is a bounded sequence in \(H^1(\Sigma _D)\), from the previous inequality and since we are assuming \(c_1=0\) we deduce that
which contradicts (6.19). Hence \(c_1>0\), and by the same argument we infer that \(c_2>0\). The proof of the claim is complete.
In order to conclude the proof of Step 2 we show that the previous claim leads to a contradiction. To this end we begin observing that by invariance by dilatation (see Remark 6.5) it follows that our minimization problem is equivalent to
In particular by (6.6) and a straightforward computation we check that
and a minimizing sequence for \({\mathcal {M}}(\Sigma _D)\) is given by \(\Lambda _k:=|{{\widetilde{\Omega }}}_{n_k} |^{-\frac{1}{N}}{{\widetilde{\Omega }}}_{n_k}\). Then, setting \(\Lambda _{i,k}:=|{{\widetilde{\Omega }}}_{n_k} |^{-\frac{1}{N}}\Omega _{i,k}\), \(i=1,2\), and in view of the previous claim, up to a subsequence, we have \(\Lambda _k= \Lambda _{1,k} \cup \Lambda _{2,k}\), where \(|\Lambda _{i,k}|\rightarrow \frac{c_i}{c_1+c_2}>0\), as \(k\rightarrow +\infty \), \(i=1,2\), and \(\Lambda _{1,k} \cap \Lambda _{2,k}=\emptyset \) for all sufficiently large k. Now, as \(\Lambda _{i,k} \subset \Sigma _D\) are quasi-open subsets with positive measure, then by definition of \({\mathcal {M}}(\Sigma _D)\), for any \(i=1,2\), we have
In addition, assuming without loss of generality that \(|\Lambda _{1,k}|\geqq |\Lambda _{2,k}|\), by an elementary computation, we deduce that
where \(\xi _k \in [0,|\Lambda _{2,k}|]\). Then, setting \({\mathcal {M}}_k(\Sigma _D):=\frac{{\mathcal {E}}(\Lambda _k; \Sigma _D)}{|\Lambda _k|^{\frac{N+2}{N}}}<0\), recalling that \((\Lambda _k)_k\) is minimizing for \({\mathcal {M}}(\Sigma _D)\), exploiting the properties of \(\Lambda _k\) and taking into account (6.23), (6.22), we infer that for all sufficiently large k it holds
where the last inequality is strict because \(|\Lambda _{i,k}|\rightarrow \frac{c_i}{c_1+c_2}>0\), for \(i=1,2\), \(k\rightarrow +\infty \), and \({\mathcal {M}}(\Sigma _D)<0\). Clearly (6.24) is contradictory and this concludes the proof of Step 2.
From Step 1 and Step 2 we know that the only admissible case is (i). Roughly speaking (i) states that, there exists a sequence \((y_{n_k})_k \subset \overline{\Sigma _D}\) such that for a sufficiently large ball \(B_R(y_{n_k})\), the mass of \(u_{n_k}\) is concentrated in \(B_R(y_{n_k})\cap \Omega _{n_k}\), while the part in the complement \(B_R^\complement (y_k)\cap \Omega _{n_k}\) is negligible. With (i) at hand we can show that the same happens for the energy, in particular the possible tails of \(\Omega _{n_k}\) do not play a role. This is the content of the next technical step.
Step 3: For any fixed \(\varepsilon >0\) there exist \({\bar{R}}>1\) and \({\bar{k}} \in {\mathbb {N}}\), both depending only on \(\varepsilon \), such that
Let us fix \(\varepsilon >0\) and let \(R>0\) be tha radius given by (i). Let \(\varphi \in C^\infty _c({\mathbb {R}}^N)\) such that \(0\leqq \varphi \leqq 1\), \(\varphi \equiv 1\) in \(B_{R}(0)\), \(\varphi \equiv 0\) in \(B_{2R}^\complement (0)\) and \(|\nabla \varphi |\leqq \frac{C_0}{R}\) in \({\mathbb {R}}^N\), where \(C_0>0\) is a constant independent of R. We set \(\varphi _k(x):=\varphi (x-y_{n_k})\) and observe that
Then, exploiting the properties of \(\varphi _k\), Hölder’s inequality and (6.16) we infer that
where \(C_0\), \(C_1\) are both independent of k and R. Similarly, for \(|\mathbf (II) |\) we have
and thus by (6.26) and assuming without loss of generality that \(R>1\) we obtain that
where \(C_2>0\) is independent of k and R. In addition, let us write
We first observe that \(\mathbf (III) =0\), because \(\varphi _k\equiv 1\) in \(B_R(y_k)\), while for (IV), applying Hölder’s inequality, taking into account that \(u_{n_k}=0\) q.e. on \(\Sigma \setminus \Omega _{n_k}\) and the properties of \(\varphi _k\), we get that
Now, thanks to (i) and (6.18) it follows that \(\Vert u_{n_k}\Vert _{L^2(\Sigma _D\cap B_R^\complement (y_k))} \leqq \sqrt{2\varepsilon }\) for all sufficiently large k, and thus, as \(|\Omega _{n_k}|\leqq c\), we deduce that
Summing up, we have proved that
Hence, combining (6.27), (6.28) and recalling the definition of the functional J (see (6.2)) we obtain
Since \(\varepsilon \) is fixed and \(C_2\) is independent of R and k, up to taking a larger R, we can assume that \(\frac{C_2}{2R}<\sqrt{2\varepsilon } c\). Then, observing that \(u_{n_k}\varphi _k \in H_0^1(B_{2R}(y_{n_k})\cap \Omega _{n_k}; \Sigma _D)\) we finally get
so that Step 3 is proved.
In the next step we prove that the sequence of points \((y_{n_k})_k\subset {\overline{\Sigma }}_D\) provided by (i) is bounded.
Step 4: The sequence \((y_{n_k})_k\subset {\overline{\Sigma }}_D\) existing by (i) is bounded.
Assume by contradiction that there exists a subsequence (still indexed by k) such that
Thanks to the assumption (6.15) we can fix \(\varepsilon >0\) sufficiently small so that
and by Step 3 we find R sufficiently large depending only on \(\varepsilon \), such that for all sufficiently large k
We observe that \(B_{2R}(y_{n_k})\cap \Omega _{n_k}\) intersects the boundary of \(\Sigma _D\). More precisely, for all sufficiently large k, it holds that
Indeed, on the contrary, setting for convenience \(\Theta _{R,k}:=B_{2R}(y_{n_k})\cap \Omega _{n_k}\) there exists a subsequence (still indexed by k) such that \({\mathcal {H}}_{N-1}(\overline{\Theta _{R,k}}\cap \partial \Sigma _D)=0\) for all \(k\in {\mathbb {N}}\), and by the same argument of [9, Remark 4.3] we conclude that \(H_0^1(\Theta _{R,k};\Sigma _D)= H_0^1(\Theta _{R,k})\), and thus
where \({\mathcal {E}}(\Theta _{R,k}; {\mathbb {R}}^N)\) denotes the “free” energy of \(\Theta _{R,k}\), under a homogeneous Dirichlet condition, namely, \({\mathcal {E}}(\Theta _{R,k}; {\mathbb {R}}^N)\) is the minimizer in \(H_0^1(\Theta _{R,k})\) of the functional \(J(v)=\frac{1}{2}\int _{{\mathbb {R}}^N} |\nabla v|^2 \ \mathrm{d}x - \int _{{\mathbb {R}}^N} v \ \mathrm{d}x\). Then, by considering the Schwartz symmetrization \(u_{\Theta _{R,k}}^*\) of the energy function \(u_{\Theta _{R,k}}\) associated to \(\Theta _{R,k}\), and thanks to the classical Pólya-Szegö inequality, we infer that
Hence, as \(\Theta _{R,k}^*\) is a ball, with \(c_k:=|\Theta _{R,k}^*|=|\Theta _{R,k}|\leqq c\), then from (6.32),(6.33), taking into account Remark 6.6 and (6.13) (noticing that \({\mathcal {E}}(\Theta _{R,k}^*; {\mathbb {R}}^N)={\mathcal {E}}(\Omega _{{\mathbb {S}}^{N-1},R_{c_k}({\mathbb {S}}^{N-1})}; \Sigma _{{\mathbb {S}}^{N-1}})\), where \(\Omega _{{\mathbb {S}}^{N-1},R_{c_k}({\mathbb {S}}^{N-1})}\) is the ball centred at the origin of radius \(R_{c_k}({\mathbb {S}}^{N-1})\) with \(|\Omega _{{\mathbb {S}}^{N-1},R_{c_k}({\mathbb {S}}^{N-1})}|=c_k\), see (6.8), (6.12)), we deduce that
Finally, recalling that \(\Theta _{R,k}=B_{2R}(y_{n_k})\cap \Omega _{n_k}\), then from (6.25) and (6.34) we have, for large k,
Hence passing to the limit as \(k\rightarrow +\infty \) we conclude that
which contradicts (6.29).
Then, by (6.31), there exists \(k_0\in {\mathbb {N}}\) such that \(\mathrm {dist}(y_{n_k}, \partial \Sigma _D)\leqq 2R\) for all \(k\geqq k_0\) and we can find a sequence of points \((z_k)_k \subset \partial \Sigma _D\setminus \{0\}\) such that \(z_k \in (\overline{B_{2R}(y_{n_k})\cap \Omega _{n_k}})\cap \partial \Sigma _D\) and \(B_{2R}(y_{n_k})\subset B_{4R}(z_k)\), for all \(k\geqq k_0\). Then, by monotonicity of the torsional energy \({\mathcal {E}}\) with respect to the set inclusion, noticing that \(H_0^1(B_{2R}(y_{n_k})\cap \Omega _{n_k}; \Sigma _D)\subset H_0^1(B_{4R}(z_k)\cap \Omega _{n_k}; \Sigma _D)\), we have
for all \(k\geqq k_0\). Clearly, by construction, \(|B_{4R}(z_k)\cap \Omega _{n_k}| \leqq c\) for all \(k\geqq k_0\) and \(|z_k|\rightarrow +\infty \), as \(k\rightarrow +\infty \). We claim that, up to a further subsequence (still indexed by k) it holds
for all sufficiently large k, where \(o(1)\rightarrow 0\) as \(k\rightarrow +\infty \).
Notice that if Claim (6.36) holds then the proof of Step 4 is complete. Indeed combining (6.30), (6.35) and (6.36), up to a subsequence, we have
for all sufficiently large k. Then, passing to the limit as \(k\rightarrow +\infty \) we get
but this contradicts (6.29).
Proof of Claim (6.36): We first observe that since \(\partial \Sigma _D\setminus \{0\}\) is a smooth hypersurface, then, for any fixed \(q\in \partial D \subset \partial \Sigma _D\setminus \{0\}\) there exists an open neighborhood V of q in \(\partial \Sigma _D\setminus \{0\}\) such that \(V-q\) is the graph over \(T_q\partial \Sigma _D\) of a smooth function \(g: U\rightarrow {\mathbb {R}}\), with \(g(0)=0\), where U is an open neighborhood of the origin in \(T_q\partial \Sigma _D\) (without loss of generality we can assume that U is a ball and g is smooth in \({\overline{U}}\)). Namely, fixing a orthonormal base \({\mathcal {B}}^\prime :=\{v_1,\ldots ,v_{N-1}\}\) of \(T_q\partial \Sigma _D\) and choosing \({\mathcal {B}}:=\{v_1,\ldots ,v_{N-1},-\nu (q)\}\) as orthonormal base of \({\mathbb {R}}^N\), where \(-\nu (q)\) is the inner unit normal of \(\partial \Sigma _D\) at q, denoting by \(x^\prime =(x_1^\prime ,\ldots ,x^\prime _{N-1})\) the coordinates of the points in \(T_q\partial \Sigma _D\) with respect to \({\mathcal {B}}^\prime \) and by \(x=(x^\prime ,x_N)\) the coordinates in \({\mathbb {R}}^N\) with respect to \({\mathcal {B}}\), we can identify
where U is the orthogonal projection of \(V-q\) onto \(T_q\partial \Sigma _D\). To be precise, if \(\varphi \) is a local parametrization centered at q, i.e. \(\varphi (0)=q\), by writing \(\varphi -q=\sum _{i=1}^{N-1} [(\varphi -q) \varvec{\cdot } v_i] v_i +(\varphi -q) \varvec{\cdot } (-\nu (q))\), and since \(\sum _{i=1}^{N-1} [(\varphi -q) \varvec{\cdot } v_i] v_i\) is a locally invertible map from an open neighborhood of the origin in \({\mathbb {R}}^{N-1}\) to an open neighborhood of the origin in \(T_q\partial \Sigma _D\cong {\mathbb {R}}^{N-1}\), then, denoting by G its local inverse and taking \(g(x^\prime ):=[(\varphi -q)\circ G(x^\prime )] \varvec{\cdot } (-\nu (q))\) we obtain the desired map. In particular, notice that since \(\frac{\partial [(\varphi -q)\circ G]}{\partial x^\prime _i}(0)\in T_q\partial \Sigma _D\) it follows that \(\frac{\partial g}{\partial x_i^\prime }(0)=0\), for any \(i=1,\ldots ,N-1\).
Now, since \(\partial \Sigma _D\) is a cone it follows that for any \(t>0\), \(T_{tq}\partial \Sigma _D=T_q\partial \Sigma _D\), \(\nu (tq)=\nu (q)\) and \(tV-tq\) is the graph over \(T_q\partial \Sigma _D\) of the map \(g_t:tU\rightarrow {\mathbb {R}}\) defined by
For any \(x^\prime \in tU\), for any \(i=1,\ldots ,N-1\), we have
where \(\xi =\xi (x^\prime ,t)\) belongs to the segment joining 0 and \(\frac{x^\prime }{t}\), and \(\nabla _{x^\prime }\) denotes the gradient with respect to \(x_1^\prime ,\ldots ,x_{N-1}^\prime \). Hence, for any fixed ball \(B_{R_1}(0)\subset T_q\partial \Sigma _D\), for all \(t>0\) sufficiently large such that \(B_{R_1}(0) \subset tU\), recalling that \(\frac{\partial g}{\partial x^\prime _i}\left( 0\right) =0\), we have
where C is independent of t, and, in particular,
Let \(C^+_{B_{R_1}(0)}\), \(E^+\left( g_t\big |_{B_{R_1}(0)}\right) \) be, respectively, the upper cylinder generated by \(B_{R_1}(0)\) and the epigraph associated to \(g_t\big |_{B_{R_1}(0)}\), namely
Then, the map \(F_t: \overline{C^+_{B_{R_1}(0)}} \rightarrow \overline{E^+\left( g_t\big |_{B_{R_1}(0)}\right) }\) defined by
is a diffeomorphism whose Jacobian matrix is of the form
where \({\mathbb {I}}_{N-1}\) is the identity matrix of order \(N-1\), \(0_{N-1}\) is the null vector in \({\mathbb {R}}^{N-1}\), T is the transposition. Notice also that \(\mathrm {Jac}(F_t)\) is independent of \(x_N\) and in view of (6.40) it holds that
Now, let us consider the sequence \((q_k)_k\subset \partial D\), where \(q_k:=\frac{z_k}{|z_k|}\), and \((z_k)_k\subset \partial \Sigma _D\setminus \{0\}\) is the sequence appearing in Claim (6.36). Since \(\partial D\) is a compact subset of \({\mathbb {S}}^{N-1}\) we deduce that, up to a subsequence (still indexed by k) it holds that \(\text {dist}_{{\mathbb {S}}^{N-1}}(q_k, {\bar{q}})\rightarrow 0\), as \(k\rightarrow +\infty \), for some \({\bar{q}}\in \partial D\), where \(\mathrm {dist}_{{\mathbb {S}}^{N-1}}\) denotes the geodesic distance in \({\mathbb {S}}^{N-1}\). Then, from the previous discussion there exist an open neighborhood \(V_1\) of \({\bar{q}}\) in \(\partial \Sigma _D\setminus \{0\}\), a convex open neighborhood \(U_1\) of the origin in \(T_{{\bar{q}}}\partial \Sigma _D\) and a smooth function \(g_1:\overline{U_1}\rightarrow {\mathbb {R}}\) such that \(g_1(0)=0\), \(\nabla _{x^\prime }g_1(0)=0\), and \(V_1-{\bar{q}}\) is the graph over \(T_{{\bar{q}}}\partial \Sigma _D\) associated to \(g_1\big |_{U_1}\), where \(x^\prime =(x_1^\prime ,\ldots ,x_{N-1}^\prime )\) are the coordinates with respect to fixed orthonormal base \(\{{\bar{v}}_1,\ldots ,{\bar{v}}_{N-1}\}\) of \(T_{{\bar{q}}}\partial \Sigma _D\). Since \(\text {dist}_{{\mathbb {S}}^{N-1}}(q_k, {\bar{q}})\rightarrow 0\), as \(k\rightarrow +\infty \), then definitely \(q_k \in V_1\), \(\Pi _{T_{{\bar{q}}}\partial \Sigma _D}(q_k-{\bar{q}})\in U_1\), where \(\Pi _{T_{{\bar{q}}}\partial \Sigma _D}:{\mathbb {R}}^N\rightarrow T_{{\bar{q}}}\partial \Sigma _D\) is the orthogonal projection onto \(T_{{\bar{q}}}\partial \Sigma _D\) and \(\Pi _{T_{{\bar{q}}}\partial \Sigma _D}(q_k-{\bar{q}})\rightarrow 0\), as \(k\rightarrow +\infty \). Let \({\bar{x}}_k^\prime :=({\bar{x}}_{1,k}^\prime ,\ldots ,{\bar{x}}_{N-1,k}^\prime )\) be the coordinates of \(\Pi _{T_{{\bar{q}}}\partial \Sigma _D}(q_k-{\bar{q}})\) and set
Then we readily check that \(V_1-q_k\) is a cartesian graph over \(T_{{\bar{q}}}\partial \Sigma _D\), associated to \(g_{1,k}:U_{1,k}\rightarrow {\mathbb {R}}\). Notice that, since \({\bar{x}}_k^\prime \rightarrow 0\), as \(k\rightarrow +\infty \), then there exists a ball \(B_{{\bar{R}}}(0)\) in \(T_{{\bar{q}}}\partial \Sigma _D\) such that \(B_{{\bar{R}}}(0)\subset U_{1,k}\) for all sufficiently large k. In particular, setting \(t_k:=|z_k|\), recalling that \(|z_k|\rightarrow +\infty \), as \(k\rightarrow +\infty \), then, \( t_kU_{1,k}\) invades \(T_{{\bar{q}}}\partial \Sigma _D\). As in (6.37) we consider the rescaled map \(h_{t_k}:t_kU_{1,k}\rightarrow {\mathbb {R}}\) defined by
and arguing as in (6.38) we have the expansion
where \(\xi _k\) belongs to the segment joining \({\bar{x}}_k^\prime \) and \(\frac{x^{\prime }}{t_k}\). Let us fix a ball \(B_{R_1}(0)\) in \(T_{{\bar{q}}}\partial \Sigma _D\), with \(R_1\) to be chosen later and independently on k, and observe that \(B_{R_1}(0)\subset t_kU_{1,k}\) for all sufficiently large k. Since \({\bar{x}}_k^\prime \rightarrow 0\), as \(k\rightarrow +\infty \), and \(\nabla _{x^\prime } g_1(0)=0\) we get that the first term in the right-hand side of (6.45) goes to zero as \(k\rightarrow +\infty \), and arguing as in (6.39) for the second term, we conclude that
Let \(F_{t_k}: \overline{C^+_{B_{R_1}(0)}} \rightarrow \overline{E^+\left( h_{t_k}\big |_{B_{R_1}(0)}\right) }\) be the diffeomorphism defined by
where \(E^+\left( h_{t_k}\big |_{B_{R_1}(0)}\right) \) is the epigraph associated to \(h_{t_k}\big |_{B_{R_1}(0)}\) (see (6.41)) and where we recall that \(x=(x^\prime ,x_N)\) are the coordinates with respect to the orthogonal base \(\{{\bar{v}}_1,\ldots ,{\bar{v}}_{N-1},-\nu ({\bar{q}})\}\).
Now, let us consider the set \(B_{4R}(z_k)\cap \Omega _{n_k}\) appearing in (6.36). Recalling that \(z_k=t_k q_k\), since \(t_k\rightarrow +\infty \), as \(k\rightarrow +\infty \), and since R is independent of k, then, for all sufficiently large k it holds
We observe that for any k we have
and we set for brevity
Notice that since \({{{\widetilde{\Omega }}}_{R,k}}\) is a uniformly bounded subset of \({\mathbb {R}}^N\) and \(t_k(V_1-q_k)\) is the cartesian graph of \(h_{t_k}:t_kU_{1,k}\rightarrow {\mathbb {R}}\) then we can choose \(R_1>0\) (independent of k) in such a way that
for all large k. Let us denote by \({\widetilde{u}}_{k}:= u_{{{\widetilde{\Omega }}}_{R,k}} \in H_0^1({{\widetilde{\Omega }}}_{R,k}; \Sigma _D-z_k)\) the energy function of \({{\widetilde{\Omega }}}_{R,k}\) and let \({\widetilde{U}}_k:={\widetilde{u}}_k\circ F_{t_k}\). Notice that by construction and thanks to (6.49), it follows that \({\tilde{U}}_k\) extends to a function \({\tilde{U}}_k \in H_0^1(F_{t_k}^{-1}({\tilde{\Omega }}_{R,k}); {\mathbb {R}}^N_+)\).
Thanks to (6.43) we have that \(\text {det}(\text {Jac}(F_{t_k}))\equiv 1\) and thus
and
Moreover, setting
which is well defined (see (6.43)) and taking into account that
we obtain that
In view of (6.46) and recalling (6.43), (6.44) we deduce that \(M_{1,k}\rightarrow 1\) as \(k\rightarrow +\infty \). Therefore, recalling that \({\widetilde{u}}_k\) is the energy function of \({{\widetilde{\Omega }}}_{R,k}=B_{4R}(z_k)\cap \Omega _{n_k}-z_k\) and thanks to (6.52) we get that \( \int _{F_{t_k}^{-1}({\tilde{\Omega }}_{R,k})} |\nabla _{x} {\tilde{U}}_k|^2 \ \mathrm{d}x\) is bounded by a uniform positive constant. Moreover, by definition and thanks to (6.50), (6.52) we have
Hence, as \({\tilde{U}}_k \in H_0^1(F_{t_k}^{-1}({\tilde{\Omega }}_{R,k}); {\mathbb {R}}^N_+)\), denoting by \(W_k:=u_{F_{t_k}^{-1}({\tilde{\Omega }}_{R,k})} \in H_0^1(F_{t_k}^{-1} ({\tilde{\Omega }}_{R,k}); {\mathbb {R}}^N_+)\) the energy function of \(F_{t_k}^{-1}({\tilde{\Omega }}_{R,k})\) then by reflection, symmetrization and taking into account (6.51) and (6.13) we infer that
Finally, since \(\int _{F_{t_k}^{-1}({\tilde{\Omega }}_{R,k})} |\nabla _{x} {\tilde{U}}_k|^2 \ \mathrm{d}x\) is uniformly bonded and \(M_{1,k}\rightarrow 1\), as \(k\rightarrow +\infty \), then from (6.47), (6.48) and (6.54), we get
for all sufficiently large k, and this proves Claim (6.36).
In the next step, we prove the pre-compactness of the sequence \((u_{n_k})_k\) in \(L^2(\Sigma _D)\).
Step 5: The sequence \((u_{n_k})_k\) admits a subsequence which strongly converges in \(L^2(\Sigma _D)\).
We first show that
Indeed, if (6.55) is not true there exist \(\varepsilon ^\prime >0\), a sequence \((R_m)_m\subset {\mathbb {R}}^+\) such that \(R_m\rightarrow +\infty \), as \(m\rightarrow +\infty \), and we find a subsequence \((n_{k_m})_m\) such that for all \(m\in {\mathbb {N}}\)
On the other hand, taking \(\varepsilon =\frac{\varepsilon ^\prime }{4}\) in (i) we find \(R^\prime >0\) depending only on \(\varepsilon ^\prime \) such that for all \(k \in {\mathbb {N}}\)
Now, in view of Step 4 we know that \((y_{n_k})_k\) is bounded, and thus there exists \(R^{\prime \prime }>0\) independent of k such that \(B_{R^\prime }(y_k)\subset B_{R^{\prime \prime }}(0)\) for all k. Hence, from (6.57) we get that
for all sufficiently large k. Finally, by writing
and recalling that \(R_m\rightarrow +\infty \), then, we have \(R_m>R^{\prime \prime }\), for all sufficiently large m, and thus from (6.56), (6.58) we deduce that
for all sufficiently large m, but this contradicts (6.18) and (6.55) is thus proved.
In order to prove the relative compactness of the sequence \((u_{n_k})_k\) in \(L^2(\Sigma _D)\), it suffices to find, for any given \(\varepsilon >0\), a relative compact sequence \((v_k)_k\) in \(L^2(\Sigma _D)\), depending on \(\varepsilon \), with the property that
Indeed, the latter property readily implies that the set \(\{u_{n_k};\ k \in {\mathbb {N}}\}\) is totally bounded in \(L^2(\Sigma _D)\), and therefore it is relative compact since \(L^2(\Sigma _D)\) is a Banach space. So let \(\varepsilon >0\). By (6.55), there exists \(R>0\) with
Hence (6.59) holds with \(v_k: = \chi _{B_R(0)}u_{n_k}\) for \(k \in {\mathbb {N}}\), where \(\chi _{B_R(0)}\) denotes the characteristic function of the ball \(B_R(0)\). Moreover, by (6.16) and the compactness of the embedding \(H^1(B_R(0)\cap \Sigma _D) \hookrightarrow L^2(B_R(0)\cap \Sigma _D)\) the sequence of functions \(u_{n_k}\big |_{B_R(0)}\), \(k \in {\mathbb {N}}\) is relatively compact in \(L^2(B_R(0)\cap \Sigma _D)\), which obviously implies that the sequence \((v_k)_k\) is relatively compact in \(L^2(\Sigma _D)\), as required. We have thus established the relative compactness of the sequence \((u_{n_k})_k\) in \(L^2(\Sigma _D)\), as claimed.
Step 6: Existence of a minimizer for \({\mathcal {O}}_c(\Sigma _D)\).
In the previous steps we proved that the sequence of energy functions \((u_n)_n\), associated to a minimizing sequence \((\Omega _n)_n \subset \Sigma _D\) for \({\mathcal {O}}_c(\Sigma _D)\), is bounded in \(H^1(\Sigma _D)\) (see (6.16)) and possesses a subsequence which strongly converges in \(L^2(\Sigma _D)\). Hence, up to a subsequence (still indexed by n for convenience), we have \(u_n \rightharpoonup {\bar{u}}\) in \(H^1(\Sigma _D)\), for some \({\bar{u}} \in H^1(\Sigma _D)\), and \(u_n \rightarrow {\bar{u}}\) in \(L^2(\Sigma _D)\), as \(n\rightarrow +\infty \).
We set \(\Omega :=\{{\bar{u}}>0\} \subset \Sigma _D\). Since \({\bar{u}}\in H^1(\Sigma _D)\) then \(\Omega \) is a quasi-open subset of \(\Sigma _D\), in addition, arguing as in [8, Proof of Lemma 5.2], namely using that \(u_n \rightarrow {\bar{u}}\) in \(L^2(\Sigma _D)\), as \(n\rightarrow +\infty \), and applying Fatou’s Lemma, we infer that
We claim that \(\Omega \) is a minimizer for \({\mathcal {O}}_c(\Sigma _D)\) and that \({\bar{u}}\) is the torsion function of \(u_\Omega \). To prove this we first observe that as \(u_n \rightarrow {\bar{u}}\) in \(L^2(\Sigma _D)\) and since \(|\Omega _n|\leqq c\), \(|\Omega |\leqq c\) it follows that \(u_n \rightarrow {\bar{u}}\) in \(L^1(\Sigma _D)\). Indeed by construction we have \(u_n\in H_0^1(\Omega _n;\Sigma _D)\), \({\bar{u}} \in H_0^1(\Omega ;\Sigma _D)\), and by Hölder’s inequality we deduce that
Now, as \(u_n \rightharpoonup {\bar{u}}\) in \(H^1(\Sigma _D)\), we have
and thus, since \(u_n \rightarrow {\bar{u}}\) in \(L^2(\Sigma _D)\), we readily get that
Then, recalling the definition of the functional J (see (6.2)), exploiting that \(u_n \rightarrow {\bar{u}}\) in \(L^1(\Sigma _D)\) and since \(u_n\) are the energy functions associated to \(\Omega _n\), we obtain that
Finally, considering the energy function \(u_{\Omega }\) associated to \(\Omega \), i.e. the minimizer of J in \(H_0^1(\Omega ;\Sigma _D)\), then, by the minimality of \(u_{\Omega }\), since \({\bar{u}} \in H_0^1(\Omega ;\Sigma _D)\) and thanks to (6.60) we have
Therefore \({\mathcal {E}}(\Omega ; \Sigma _D)={\mathcal {O}}_c(\Sigma _D)\), and (6.61) implies that \(J(u_\Omega )=J({\bar{u}})\). Hence \(\Omega \) is a minimizer for \({\mathcal {O}}_c(\Sigma _D)\) and \({\bar{u}}=u_\Omega \) in \(H^1(\Sigma _D)\). \(\square \)
Corollary 6.9
If \(D\subset {\mathbb {S}}^{N-1}\) is a smooth domain such that
then \(\mathcal {O}_c(\Sigma _D)\) is achieved, for any \(c>0\).
Proof
By (6.9)–(6.13) we readily check that (6.62) implies the condition (6.15), and by Theorem 6.8 we conclude. \(\square \)
We conclude this section with
Proposition 6.10
Let \(D\subset {\mathbb {S}}^{N-1}\) be a smooth domain and let \(c>0\). Then
Proof
Let us fix \(q\in \partial D\subset \partial \Sigma _D\setminus \{0\}\) and let \(\{v_1,\ldots ,v_{N-1}\}\) be an orthonormal basis of \(T_q\partial \Sigma _D\). We denote by \(x=(x^\prime ,x_N)\) the coordinates of points in \({\mathbb {R}}^N\) with respect to \(\{v_1,\ldots ,v_{N-1},-\nu (q)\}\), where \(-\nu (q)\) is the inner unit normal to \(\partial \Sigma _D\) at q. As seen in the proof of Claim (6.36) there exist an open neighborhood V of q in \(\partial \Sigma _D\setminus \{0\}\), an open neighborhood U of the origin in \(T_q\partial \Sigma _D\), and a smooth map \(g:{\overline{U}}\rightarrow {\mathbb {R}}\), \(g=g(x^\prime )\) such that \(V-q\) is the graph over \(T_q\partial \Sigma _D\) of \(g\big |_{U}\).
Let \(B^+_{R}(0)\subset {\mathbb {R}}^N_+\) be a N-dimensional half-ball such that \(|B^+_{R}(0)|=c\), i.e. \(B^+_{R}(0)\) is a half-ball of volume c contained in the upper half-space delimited by \(T_q\partial \Sigma _D\). Let \(u_{B^+_{R}(0)}\in H_0^1(B^+_{R}(0); {\mathbb {R}}^N_+)\) be the energy function of \(B^+_{R}(0)\). Then, by definition and recalling Remark 6.7, we have
Let \(B_{R_1}(0)\) be a ball in \(T_q\partial \Sigma _D\), with \(R_1>R\). Clearly
Let \((t_k)_k\subset {\mathbb {R}}^+\) be a sequence such that \(t_k\rightarrow +\infty \), as \(k\rightarrow +\infty \), then, setting \(z_k:=t_k q\) we obtain a diverging sequence of points on \(\partial \Sigma _D\setminus \{0\}\). We consider the rescaled map \(g_{t_k}:t_k U\rightarrow {\mathbb {R}}\) defined by (6.37) and the associated diffeomorphism \(F_{t_k}: \overline{C^+_{B_{R_1}(0)}} \rightarrow \overline{E^+\left( g_{t_k}\big |_{B_{R_1}(0)}\right) }\) given by (6.42), where \(E^+\left( g_{t_k}\big |_{B_{R_1}(0)}\right) \), \(C^+_{B_{R_1}(0)}\) are defined by (6.41). The inverse diffeomorphism \(F_{t_k}^{-1}\) is given by
and as done in (6.43), (6.44) we readily check that
Moreover, setting \(U_k:=u_{B^+_R(0)}\circ F_{t_k}^{-1}\) we notice that since \(u_{B^+_R(0)}\in H_0^1(B^+_{R}(0); {\mathbb {R}}^N_+)\) (actually \(u_{B^+_R(0)}= 0\) in \({\mathbb {R}}^N_+\setminus B^+_{R}(0)\), see (6.9) with \(D={\mathbb {S}}^{N-1}_+\)), then, by construction, taking into account (6.65), it follows that \(U_k\) extends to a function \(U_k\in H_0^1(F_{t_k}\left( B^+_{R}(0)\right) ; \Sigma _D-z_k)\). Arguing as in (6.50)–(6.52), taking into account (6.67), we infer that \(|F_{t_k}\left( B^+_{R}(0)\right) |=|B^+_{R}(0)|=c\),
where
and \(M_{2,k}\rightarrow 1\), as \(k\rightarrow +\infty \). Hence, combining (6.64), (6.68) and (6.69) we deduce
where in the last inequality we used that \(U_k\in H_0^1(F_{t_k}\left( B^+_{R}(0)\right) ; \Sigma _D-z_k)\), the definition of torsional energy of \(F_{t_k}\left( B^+_{R}(0)\right) \), \(M_{2,k}\rightarrow 1\), as \(k\rightarrow +\infty \) and that \(\int _{F_{t_k}\left( B^+_{R}(0)\right) } |\nabla _{x} U_k|^2 \ \mathrm{d}x\) is uniformly bounded. Summing up, from (6.70) and the definition of \(\mathcal {O}_c(\Sigma _D)\) we finally have
and passing to the limit as \(k\rightarrow +\infty \) we obtain (6.63). \(\square \)
7 Properties of Minimizers and Proof of Theorem 1.2
In this section we show some qualitative properties of the minimizers of the torsional energy functional with fixed volume (we refer to Sect. 6 for the notations). In view of the scaling invariance of our problem (see Remark 6.5) it suffices to focus on the case \(\mathcal {O}_1(\Sigma _D)\). We begin by proving that any minimizer for \(\mathcal {O}_1(\Sigma _D)\) is bounded.
Proposition 7.1
If \(\Omega \) is a minimizer for \(\mathcal {O}_1(\Sigma _D)\) then \(\Omega \) is bounded.
Proof
We argue as in [3, Sect. 2.1.2] with slightly changes. Let \(\Omega \) be a minimizer for \(\mathcal {O}_1(\Sigma _D)\). In view of [7, Theorem 1], in order to prove that \(\Omega \) is bounded it is sufficient to show that \(\Omega \) is a local shape subsolution for the energy \({{\mathcal {T}}}_D\), which means that, there exist \(\delta >0\) and \(\Lambda >0\) such that for any quasi-open subset \({{\widetilde{\Omega }}}\subset \Omega \) with \(\Vert u_{\Omega }-u_{{{\widetilde{\Omega }}}}\Vert _{L^2{\Sigma _D}}< \delta \), it holds that
Let us assume, by contradiction, that there exist a sequence \((\Lambda _n)_n\subset {\mathbb {R}}^+\) with \(\Lambda _n\rightarrow 0\), as \(n\rightarrow +\infty \), and an increasing sequence \(({{\widetilde{\Omega }}}_n)_n\subset \Omega \) of quasi-open subsets such that
and \(\Vert u_{\Omega }-u_{{{\widetilde{\Omega }}}_n}\Vert _{L^2{\Sigma _D}}\rightarrow 0\), as \(n\rightarrow +\infty \). Then, let us fix \(t_n>1\) such that
Obviously, \(t_n\rightarrow 1^+\), as \(n\rightarrow +\infty \), and by the minimality of \(\Omega \) we have
Thus, from (7.1) we obtain
and dividing by \(t_n^N-1=\frac{|\Omega |}{|{{\widetilde{\Omega }}}_n|}-1\) we get
This gives a contradiction because, as \(n\rightarrow +\infty \), the left-hand side of (7.2) converges to \(-\frac{N+2}{N}{\mathcal {E}}(\Omega ; \Sigma _D)\), while the right-hand side converges to zero. \(\square \)
Proposition 7.2
If \(\Omega \) is a minimizer for \(\mathcal {O}_1(\Sigma _D)\) then the torsion function \(u_\Omega \in H_0^1(\Omega ; \Sigma _D)\) is Lipschitz continuous in any Lipschitz domain \(\omega \subset \Sigma _D\) such that \({\overline{\omega }}\subset \Sigma _D\) and \(\Omega =\{u_\Omega >0\}\) is an open subset of \(\Sigma _D\).
Proof
The result follows essentially as in the case of Dirichlet boundary conditions, which was addressed in the work [5]. The extension to the case of mixed boundary conditions was done in [21, Theorem 2.14] for the problem of minimizing the first eigenvalue. In our situation the proof would be similar. \(\square \)
Next, we prove that any minimizer is connected.
Proposition 7.3
If \(\Omega \) is a minimizer for \(\mathcal {O}_1(\Sigma _D)\) then \(\Omega \) is a connected subset of \(\Sigma _D\).
Proof
As seen in the proof of Theorem 6.8, Step 2, we notice that, as \(|\Omega |=1\), then \(\Omega \) is also a minimizer for
Assume by contradiction that \(\Omega \) is not connected. Then there exist two open subsets \(\Omega _1\), \(\Omega _2\) of \(\Sigma _D\), with \(\Omega _1, \Omega _2\ne \emptyset \), such that \(\Omega _1\cap \Omega _2=\emptyset \) and
In addition, since \(\Omega _i\) is an open nonempty subset of \(\Sigma _D\) then \(|\Omega _i|>0\), for \(i=1,2\), and by construction we have
Then, since \(\frac{N+2}{N}>1\), by the convexity of \(t\mapsto t^{\frac{N+2}{N}}\), taking into account that \(|\Omega _1|>0\), \(|\Omega _2|>0\) and \({\mathcal {M}}(\Sigma _D)<0\), we deduce that
which is a contradiction. \(\square \)
Concerning the regularity of the minimizing set, by the theory of free boundary problems we have
Proposition 7.4
Let \(\Omega \subset \Sigma _D\) be a minimizer for \(\mathcal {O}_1(\Sigma _D)\) and let \(\Gamma =\partial \Omega \cap \Sigma _D\) be its relative boundary. Then there exists a critical dimension \(d^*\) which can be either 5,6 or 7, such that
-
(i)
\(\Gamma \) is smooth if \(N<d^*\);
-
(ii)
\(\Gamma \) can have countable isolated singularities if \(N=d^*\);
-
(iii)
\(\Gamma \) can have a singular set of dimension \(N-d^*\), if \(N>d^*\).
Moreover on the regular part of \(\Gamma \) the normal derivative \(\frac{\partial u_\Omega }{\partial \nu }\) is constant, namely \(\frac{\partial u_\Omega }{\partial \nu }\equiv - \sqrt{\frac{2(N+2)}{N} |{\mathcal {O}}_1(\Sigma _D)|}\), where \(u_\Omega \) is the torsion function of \(\Omega \).
Proof
The points (i)–(iii) follows from the results of [11, 15] and [27]. Let us prove the last statement.
Let \(\Gamma _{reg}\) be the regular part of \(\Gamma \) (which is a relative open set of \(\Gamma \)), let \(x_0\in \Gamma _{reg}\), and let \(B_r(x_0)\) be a small ball such that \({B_{2r}(x_0)}\subset \Sigma _D\) and \(\Gamma \cap {B_{2r}(x_0)} \subset \Gamma _{reg}\). Moreover, let \(\psi \in C^\infty _c(B_r(x_0))\) and consider the vector field V given by \(V=\psi {\bar{\nu }}\), where \({\bar{\nu }}\) is a smooth extension of the normal versor \(\left. \nu \right| _{\Gamma \cap {B_{2r}(x_0)}}\) of \(\Gamma \cap {B_{2r}(x_0)}\) to a smooth vector field defined in \(\overline{B_r(x_0)}\). Hence, by construction, we have that \(V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}^N\) is a smooth vector field with compact support in \(B_r(x_0)\), and in particular it holds that \(V(0)=0\) and \(V(x)=0\in T_x\partial \Sigma _D\) for all \(x\in \partial \Sigma _D\setminus \{0\}\). This means that the associated flow \(\xi :(-t_0,t_0)\times \overline{\Sigma _D}\rightarrow \overline{\Sigma _D}\), for some \(t_0>0\), preserves the boundary \(\partial \Sigma _D\) and we consider the induced deformation of \(\Omega \), \((\Omega _t)_{t\in (-t_0,t_0)}\), where \(\Omega _t:=\xi (t,\Omega )\). Actually, since \(\mathrm {supp}(\psi ) \subset B_r(x_0)\) we infer that \(\xi (t,x)=x\) for \(x \in B_{\frac{3}{2} r}^\complement (x_0)\), \(t\in (-t_0,t_0)\).
Let \(u_{\Omega _t}\in H^1_0(\Omega _t;\Sigma _D)\) be the torsion function of \(\Omega _t\), for \(t\in (-t_0,t_0)\) and let us set \(u_t:=u_{\Omega _t}\). Arguing as in the proof of [23, Proposition 4.3] we can prove that the map from \((-t_0,t_0)\) to \(H^1(\Sigma _D)\), \(t\mapsto u_t\) is differentiable. In particular the function \(f:(-t_0,t_0)\rightarrow L^1(\Sigma _D)\), given by \(f(t)=|\nabla u_t|^2\) is differentiable. We also notice that, since \(u_\Omega \) is a weak solution to (6.1), then by standard elliptic regularity theory \(u_\Omega \in W^{2,2}(\Omega \cap B_r(x_0))\). In particular it holds that \(V f(0)=\psi {\bar{\nu }}|\nabla u_\Omega |^2\in W^{1,1}(\Sigma _D,{\mathbb {R}}^N)\) and by easy modifications to the proof of [14, Theorem 5.2.2] we infer that the function \(t\mapsto {\mathcal {E}}(\Omega _t;\Sigma _D)= -\frac{1}{2}\int _{\Omega _t}|\nabla u_t|^2 \ \mathrm{d}x\) is differentiable at \(t=0\) and
where \(u^\prime =\frac{\mathrm{d}}{\mathrm{d}t}\left. \left( u_t\right) \right| _{t=0}\) is a solution to (see the proof of [23, Proposition 4.3])
We point out that since the flow \(\xi \) leaves invariant \(B^\complement _{\frac{3}{2}r}(x_0)\) for all \(t\in (-t_0,t_0)\), we have \(u^\prime \equiv 0\) in \(\Omega \cap B^\complement _{\frac{3}{2}r}(x_0)\) and thus
Let us analyse \(\mathbf (I) \). We first observe that as \(\Gamma \cap B_{2r}(x_0)\subset \Gamma _{reg}\) and \(u^\prime \) is a solution to (7.3), then by standard elliptic regularity theory, it follows that \(u^\prime \in W^{2,2}\left( \Omega \cap B_{\frac{3}{2}r}(x_0)\right) \) and it is smooth inside \(\Omega \). Hence, applying the Green’s formula, taking into account that \(\Delta u^\prime =0\) in \(\Omega \cap B_{\frac{3}{2}r}(x_0)\), \(\frac{\partial u^\prime }{\partial \nu }=0\) on \(\Omega \cap \partial B_{\frac{3}{2}r}(x_0)\) (because \(u^\prime \equiv 0\) in \(B_{\frac{3}{2} r}^\complement (x_0)\) and \(u^\prime \) is smooth inside \(\Omega \)) and \(u_\Omega =0\) on \(\Gamma \cap \overline{B_{\frac{3}{2}r}(x_0)}\) we get that \(\mathbf (I) =0\). For \(\mathbf (II) \), applying the divergence theorem, and recalling the definition of V, in the end, we obtain
On the other hand, for the volume, we have
Now, since \(\Omega \) is a minimizer for \(\mathcal {O}_1(\Sigma _D)\), then, recalling Remark 6.5 and as observed in the proof of Step 2 of Theorem 6.8 we get that \(\Omega \) is also a minimizer for \(\frac{{\mathcal {E}}(\Omega ;\Sigma _D)}{|\Omega |^{\frac{N+2}{N}}}\). Thus from (7.5), (7.6), since \(|\Omega |=1\) and \({\mathcal {E}}(\Omega ;\Sigma _D)= {\mathcal {O}}_1(\Sigma _D)<0\), we readily obtain that
Finally, from the arbitrariness of \(\psi \) and \(x_0\) we conclude that \(|\nabla u_\Omega |^2\equiv \frac{2(N+2)}{N} |{\mathcal {O}}_1(\Sigma _D)|\) on \(\Gamma _{reg}\), and as \(u_\Omega =0\) on \(\Gamma _{reg}\) then by Hopf’s lemma it follows that \(\frac{\partial u_\Omega }{\partial \nu }\equiv - \sqrt{\frac{2(N+2)}{N} |{\mathcal {O}}_1(\Sigma _D)|}\) on \(\Gamma _{reg}\). \(\square \)
We conclude this section with the following:
Proof of Theorem 1.2
It follows from Theorem 5.1, Theorem 6.8, Corollary 6.9 and Proposition 7.4. \(\square \)
8 The Isoperimetric Problem and Proof of Theorem 1.3
In this section we study the isoperimetric problem in the class of strictly star-shaped domains in cones, i.e. domains in \(\Sigma _D\) whose relative boundary is a radial graph.
Using the same notations of the previous sections, if \(\varphi \in C^2({{\overline{D}}}, {\mathbb {R}})\) and \(\Omega _\varphi \), \(\Gamma _\varphi \) are, respectively, the associated star-shaped domain (see (3.3)), and the associated radial graph (see Definition 2.1), the (relative) perimeter of \(\Omega _\varphi \) in \(\Sigma _D\) is given by
where \(\mathrm{d}\sigma \) is the \((N-1)\)-dimensional area element of \({\mathbb {S}}^{N-1}\). Let us observe that \({\mathcal {P}}\) can be seen as a functional on the space \(C^2({{\overline{D}}}, {\mathbb {R}})\) and, contrarily to the torsional energy functional of Sect. 3, its expression does not involve the associated domain \(\Omega _\varphi \). Thus we set for brevity \({\mathcal {P}}(\varphi )={\mathcal {P}}(\Omega _\varphi ; \Sigma _D)\).
The derivative of \({\mathcal {P}}\) along a variation \(v\in C^2({{\overline{D}}}, {\mathbb {R}})\) is given by
Let \({\mathcal {V}}\) be the volume functional (see (4.1)). We are concerned with critical points \(\varphi \) of \({\mathcal {P}}\) subject to the volume constraint \(\{{\mathcal {V}}=c\}\). Namely we consider the manifold M defined by (4.4) and the restriction \({\mathcal {I}}:=\left. {\mathcal {P}}\right| _M\). A critical point \(\varphi \in M\) for \({\mathcal {I}}\) satisfies
with a Lagrangian multiplier \(\uplambda \in {\mathbb {R}}\). In the next two propositions we prove that the radial graph \(\Gamma _\varphi \) associated to a critical point \(\varphi \in M\) is a CMC hypersurface which intersects orthogonally \(\partial \Sigma _D\setminus \{0\}\).
This is well known if the variations of the domains are taken in the whole class of subsets of \(\Sigma _D\) of finite relative perimeter (see [25]) but not obvious in our case.
Proposition 8.1
If \(\varphi \in C^2({{\overline{D}}},{\mathbb {R}})\) is a volume-constrained critical point for \({\mathcal {P}}\), then the associated radial graph \(\Gamma _\varphi \) has constant mean curvature \(H\equiv \frac{\uplambda }{N-1}\).
Proof
Let \(v \in C^{1}_c({D})\) be a variation with compact support. By definition (see (8.1)) there exists \(\uplambda \in {\mathbb {R}}\) such that
Let us observe that
Hence we can rewrite (8.2) as
Now, since \(\varphi \) is smooth and v has compact support in D, integrating by parts, we get
and thus we deduce
Therefore, as v is arbitrary, we obtain
i.e.
Comparing (8.4) with (2.10) it follows that the mean curvature of \(\Gamma _\varphi \) is constant and it is equal to \(\frac{\uplambda }{N-1}\). \(\square \)
Proposition 8.2
If \(\varphi \) is as in the statement of Proposition 8.1 then \(\Gamma _\varphi \) intersects orthogonally \(\partial \Sigma _D\setminus \{0\}\).
Proof
Let \(\nu _{_{\partial \Sigma _D}}\) be the exterior unit normal to \(\partial \Sigma _D\setminus \{0\}\), and let \(\nu _{_{\Gamma _\varphi }}\) be the exterior unit normal to \(\Gamma _\varphi \). By (2.7) we have
where \({\mathcal {Y}}\) is the standard parametrization of \(\Gamma _\varphi \) defined by (2.5). Notice that, since by assumption \(\varphi \) is smooth up to the boundary, then (8.5) is well defined on \({{\overline{D}}}\). If \(p \in (\partial \Sigma _D\setminus \{0\}) \cap \overline{\Gamma _\varphi }\), then by definition the intersection is orthogonal at p if and only if \(\nu _{_{\partial \Sigma _D}}(p) \varvec{\cdot } \nu _{_{\Gamma _\varphi }}(p) = 0\). Therefore, writing \(p={\mathcal {Y}}(q)\) this is equivalent to
Since \(p={\mathcal {Y}}(q) \in \partial \Sigma _D\setminus \{0\}\) and \(\partial \Sigma _D\setminus \{0\}\) is the boundary of a cone we have \(\nu _{_{\partial \Sigma _D}}({\mathcal {Y}}(q)) \varvec{\cdot } q = 0\) and thus the intersection between \(\partial \Sigma _D\setminus \{0\}\) and \(\Gamma _\varphi \) is orthogonal if and only if
Exploiting again that \(\partial \Sigma _D\) is a cone, we have \(\nu _{_{\partial \Sigma _D}}(p)=\nu _{_{\partial \Sigma _D}}(t p)\) for any \(p\in \partial \Sigma _D\setminus \{0\}\), \(t>0\). Hence, since \({\mathcal {Y}}(q) \in \partial \Sigma _D\setminus \{0\}\), we have \(\nu _{_{\partial \Sigma _D}}({\mathcal {Y}}(q))=\nu _{_{\partial \Sigma _D}}(q)=\nu _{_{\partial D}}(q)\) for any \(q \in \partial D\), where \(\nu _{_{\partial D}}\) is the exterior unit co-normal to \(\partial D\), and thus (8.6) is equivalent to
To prove this, we argue as in the proof of Proposition 8.1. Taking a variation \(v \in C^1({{\overline{D}}}, {\mathbb {R}})\) and integrating by parts we have
Using this and arguing as in the proof Proposition 8.1, since \(\varphi \) satisfies the equation (8.4) we obtain
Since \(v \in C^1({{\overline{D}}}, {\mathbb {R}})\) is arbitrary we can choose v such that \(v= \ \left\langle \frac{\nabla \varphi }{ \sqrt{1+|\nabla \varphi |^2}}, \nu _{_{\partial D}} \right\rangle \) on \(\partial D\) and thus
which gives \(\left\langle \frac{\nabla \varphi }{ \sqrt{1+|\nabla \varphi |^2}}, \nu _{_{\partial D}} \right\rangle \equiv 0\) on \(\partial D\), and thus (8.7) is proved. \(\square \)
Analogously to Lemma 4.3, if \(\varphi \in M\) is a critical point for \({\mathcal {I}}\) then
Choosing \(c=|\Omega _0|=|\Sigma _D\cap B_1(0)|\) in (4.4) we observe that the function \(\varphi \equiv 0\) belongs to M and it is a critical point for \({\mathcal {I}}\). In particular (8.1) yields \(\uplambda =N-1\). Moreover, for any \(v,w \in T_0M\), since
and recalling (4.3), it follows that
From (8.8) we easily have the analogue of Theorem 5.1 for the perimeter functional \({\mathcal {I}}\).
Theorem 8.3
Let \(\uplambda _1(D)\) be the first nontrivial eigenvalue of \(-\Delta _{{\mathbb {S}}^{N-1}}\) on the domain D with zero Neumann condition on \(\partial D\). Then
-
(i)
if \(\uplambda _1(D) < N-1\) then \(\varphi \equiv 0\) is not a local minimizer for \({\mathcal {I}}\);
-
(ii)
if \(\uplambda _1(D) > N-1\) then \(\varphi \equiv 0\) is a local minimizer for \({\mathcal {I}}\).
Proof
Since \(T_0M\) is made by functions with zero mean value (see (4.5)), considering the \(L^2\)-normalized eigenfunction \(w_1\) corresponding to the eigenvalue \(\uplambda _1(D)\), from (8.8) we get \({\mathcal {I}}^{\prime \prime }(0)[w_1,w_1]<0\) whenever \(\uplambda _1(D)<N-1\). This proves (i).
Viceversa, if \(\uplambda _1(D)>N-1\), from (8.8) and the variational characterization of \(\uplambda _1(D)\) we get that \({\mathcal {I}}^{\prime \prime }[v,v]>0\) for all \(v\in T_0M\) with \(v\ne 0\), and hence (ii) holds. \(\square \)
To find examples of domains \(D\subset {\mathbb {S}}^{N-1}\) satisfying \(\uplambda _1(D)<N-1\) we can use the function \(u_e\in C^\infty ({\mathbb {S}}^{N-1})\) introduced in (5.8) and Proposition 5.2. Hence for the nonconvex domains constructed in the “Appendix”, the spherical sectors are not the minimizers of \({\mathcal {I}}\).
Concerning the existence of a minimizer for the relative perimeter \({\mathcal {P}}(E;\Sigma _D)\) in the whole class of finite perimeter subsets of \(\Sigma _D\), with a fixed volume, we summarise in the following the results stated in [25].
Theorem 8.4
Let \(D\subset {\mathbb {S}}^{N-1}\) be a domain such that \({\mathcal {H}}_{N-1}(D)\leqq {\mathcal {H}}_{N-1}({\mathbb {S}}^{N-1}_+)\). Then, there exists a set of finite perimeter \(E^*\) inside \(\Sigma _D\) which minimizes the relative perimeter under a volume constraint, for any value of the volume. Moreover any minimizer of the relative perimeter, with fixed volume, is a bounded set.
Proof
It follows from Proposition 3.5 and Proposition 3.7 in [25]. \(\square \)
We conclude this section with the following:
Proof of Theorem 1.3
The existence of a set of finite perimeter \(E^*\) inside \(\Sigma _D\) which minimizes the relative perimeter under a volume constraint, and its boundedness, follows from Theorem 8.4. From Theorem 8.3 we infer that \(E^*\) cannot be a spherical sector, while the properties (i)-(iii) of \(\Gamma _{E^*}\) derive from classical results for isoperimetric problems (see e.g. [25, Sect. 2] and the references therein). \(\square \)
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Acknowledgements
Research partially supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)
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Appendix A. Examples of Non Convex Domains Satisfying Condition (1.7)
Appendix A. Examples of Non Convex Domains Satisfying Condition (1.7)
In this section we construct two classes of non-convex domains of \(\ {\mathbb {S}}^{N-1}\) satisfying hypothesis (1.7) of Theorem 1.2. In particular to show the instability condition \(\uplambda _1(D)<N-1\) we will prove the inequality (i) of Proposition 5.2. We begin with some preliminary technical results.
Let \(N\geqq 3\), let \(\{{e}_1,\ldots , {e}_N\}\) be the standard basis in \({\mathbb {R}}^N\), fix \(\theta \in (-\frac{\pi }{2},\frac{\pi }{2})\) and consider the vector \({e}_{\theta }\in {\mathbb {S}}^{N-1}\) defined by
By construction the vector \({e}_{\theta }\) lies in the sector \(\{ x_1>0, \ x_N>0\}\) if \(\theta \in (0,\frac{\pi }{2})\), in \(\{x_1<0, \ x_N>0\}\) if \(\theta \in (-\frac{\pi }{2},0)\), and coincides with \( {e}_{N}\) if \(\theta =0\). For any given \(r\in (0,1)\) we consider the hyperplane \(H_{\theta ,r}\) orthogonal to \({e}_{\theta }\) and passing through \(r {e}_{\theta }\), i.e.
Let \(D_{\theta ,r}\) be the region of \({\mathbb {S}}^{N-1}\) above \(H_{\theta ,r}\), namely \(D_{\theta ,r}\) is the spherical cap given by
By definition it is easy to check that \(D_{\theta ,r}\) has angular radius \(\arctan \left( \frac{\sqrt{1-r^2}}{r}\right) \). We consider the function \(u_{{e_1}}\) defined in (5.8), namely \(u_{{e_1}}(x)=x \varvec{\cdot } {e}_1=x_1\). The first result we prove is an explicit formula for the boundary integral appearing in Proposition 5.2 applied to the function \(u_{{e_1}}\) and to the domain \(D_{\theta ,r}\).
Lemma A.1
For \(\theta \in (-\frac{\pi }{2},\frac{\pi }{2})\) and \(r\in (0,1)\) it holds that
where \(\nu \) is the exterior unit co-normal (i.e., for any \(x\in \partial D_{\theta ,r}\), \(\nu (x)\) is the unique unit vector in \(T_x{\mathbb {S}}^{N-1}\) which is orthogonal to \(T_x\partial D_{\theta ,r}\) and pointing outward \(D_{\theta ,r}\)), \(\mathrm{d}{\hat{\sigma }}\) is the \((N-2)\)-dimensional area element of \({\partial D_{\theta ,r}}\), \(c_N\) is a positive constant depending only on N which is explicit (see (A.13)).
Proof
Let \(\theta \in (-\frac{\pi }{2},\frac{\pi }{2})\), \(r\in (0,1)\). We begin observing that for any \(x\in \partial D_{\theta ,r}\) it holds that
Indeed, \(x \varvec{\cdot } {e}_\theta =r\) and thus \((rx-{e}_\theta ) \varvec{\cdot } (rx-{e}_\theta )=1-r^2\), namely \(\left| \frac{1}{\sqrt{1-r^2}} (rx-{e}_\theta )\right| =1\), and we readily check that \(\frac{1}{\sqrt{1-r^2}} (rx-{e}_\theta ) \varvec{\cdot } x= \frac{1}{\sqrt{1-r^2}} (r-x \varvec{\cdot } {e}_\theta )=0\), which means that \(\frac{1}{\sqrt{1-r^2}} (rx-{e}_\theta )\in T_x{\mathbb {S}}^{N-1}\). In addition, if \(v\in T_x\partial D_{\theta ,r}\) and \(\gamma :(-\delta ,\delta )\rightarrow \partial D_{\theta ,r}\) is curve such that \(\gamma (0)=x\) and \(\gamma ^\prime (0)=v\), for some small \(\delta >0\), then as \( \partial D_{\theta ,r}\subset {\mathbb {S}}^{N-1}\), differentiating the identity \(|\gamma |^2\equiv 1\) we get \(x \varvec{\cdot } v=0\). Similarly, differentiating the identity \(\gamma \varvec{\cdot } {e}_\theta \equiv r\) we obtain \(v \varvec{\cdot } {e}_\theta =0\). Hence, \(\frac{1}{\sqrt{1-r^2}} (rx-{e}_\theta ) \varvec{\cdot } v=0\) and thus from the arbitrariness of \(v\in T_x\partial D_{\theta ,r}\) we infer that \(\frac{1}{\sqrt{1-r^2}} (rx-{e}_\theta ) \perp T_x\partial D_{\theta ,r}\), so that \(\nu (x)=\pm \frac{1}{\sqrt{1-r^2}} (rx-{e}_\theta )\). To choose the right sign we now show that \(\frac{1}{\sqrt{1-r^2}} (rx-{e}_\theta )\) points outward \(D_{\theta ,r}\). To this end, let \(x\in \partial D_{\theta ,r}\) and consider the vector \(y_s:=x+ \frac{s}{\sqrt{1-r^2}} (rx-{e}_\theta )\), with |s| small. As \(r\in (0,1)\) and since \(y_s \varvec{\cdot } {e}_\theta = r + \frac{s(r^2-1)}{\sqrt{1-r^2}}\), then, using the definition (A.1) and by elementary computations we readily check that \(\frac{y_s}{|y_s|} \in D_{\theta ,r}\) if and only if \(s<0\) with |s| is small enough, and this proves that \(\frac{1}{\sqrt{1-r^2}} (rx-{e}_\theta )\) points outward \(D_{\theta ,r}\).
With (A.3) at hand, and recalling the definition of \(u_{{e}_1}\) we easily obtain that
for any \(x=(x_1,\ldots ,x_N)\in \partial D_{\theta ,r}\). In order to compute the integral in (A.2) we determine a suitable parametrization for \(\partial D_{\theta ,r}\). To this aim, we recall that \(x=(x_1,\ldots ,x_N)\in \partial D_{\theta ,r}\) if and only if x satisfies
and by elementary computations we find that
Using the spherical coordinates for \(\ {\mathbb {S}}^{N-2}\subset {\mathbb {R}}^{N-1}\) and the second equation in (A.5) we find a parametrization for \(\partial D_{\theta ,r}\). Indeed, first assuming that \(N\geqq 4\) and denoting by \(\phi _1,\ldots ,\phi _{N-2}\) the angular coordinates, where \(\phi _1,\ldots ,\phi _{N-3}\) have the range \((0,\pi )\) and \(\phi _{N-2}\) ranges over \((0,2\pi )\), we easily check that
is a parametrization of the ellipsoid in \({\mathbb {R}}^{N-1}\) (or the sphere if \(\theta =0\)) described by the first equation in (A.5) (up to a zero measure set with respect to the \((N-2)\)-dimensional Haussdorf measure). Hence, exploiting the second equation in (A.5), we deduce that \(\psi :(0,\pi )\times \ldots \times (0,\pi )\times (0,2\pi )\rightarrow {\mathbb {R}}^N\) defined by
is a parametrization of the \((N-2)\)-dimensional sphere \(\partial D_{\theta ,r}\). Then, arguing by induction and after a straightforward computation we see that the coefficients \(g_{ij}=\left\langle \frac{\partial \psi }{\partial \phi _i}, \frac{\partial \psi }{\partial \phi _j}\right\rangle \), \(i,j=1,\ldots ,N-2\), of the induced metric on \(\partial D_{\theta ,r}\), are given by
In particular, the matrix \((g_{ij})_{i,j=1,\ldots ,N-2}\) is diagonal, positive definite and the square root of its determinant is given by
Therefore, when \(N\geqq 6\) (the other cases \(N=4,5\) being similar and easier) the \((N-2)\)-dimensional area element of \(\partial D_{\theta ,r}\) is expressed in these local coordinates by
Setting for brevity \(G(\phi _2,\ldots ,\phi _{N-2}):=(\sin \phi _2)^{N-4}\cdots \sin \phi _{N-3}\), \(\mathrm{d}\Phi :=\mathrm{d}\phi _{2}\cdots \mathrm{d}\phi _{N-3} \mathrm{d}\phi _{N-2}\), and observing that
where \(\omega _{N-2}\) is the volume of the unit ball in \({\mathbb {R}}^{N-2}\), then, recalling (A.4), exploiting Fubini’s theorem and taking into account (A.7), (A.8) we get that
where in the last integral we have discarded the linear terms in \(\cos \phi _1\) because
Rearranging the terms in the last integral of (A.9), we obtain
Now, let us observe that, for any \(k\in {\mathbb {N}}\), \(k\geqq 2\), integrating by parts, we have
Combining (A.10) and (A.11) (with \(k=N-3\)) we get that
and this proves (A.2), with \(c_N\) given by
The proof of the Lemma is then complete for \(N\geqq 4\).
When \(N=3\) the proof is much simpler. Indeed, in this case, a parametrization of the circle \(\partial D_{\theta ,r}\) is given by the map \(\psi :(0,2\pi )\rightarrow {\mathbb {R}}^3\) defined by
Hence, using the definition, (A.4), and taking into account that \(|\psi ^\prime (\phi )|=\sqrt{1-r^2}\), we easily check that
In particular, as \(\omega _1=2\), (A.12) holds true with \(c_N\) given by (A.13) even for \(N=3\). The proof is complete. \(\square \)
Remark A.2
Let \(\theta \in (0,\frac{\pi }{2})\). From the analytic expression of \(\partial D_{\theta ,r}\) given by (A.5) or, equivalently, by using (A.7), but with \(\phi _1\) varying in \([0,2\pi )\), it is easy to check that \(\partial D_{\theta ,r}\) is contained in the sector \(\{x\in {\mathbb {S}}^{N-1}; \ x_1>0, \ x_N>0\}\) if \(-\sqrt{1-r^2} \cos \theta + r \sin \theta >0\) and \(-\sqrt{1-r^2} \sin \theta + r \cos \theta >0\), which both hold true if \(r>\sqrt{\max \{\cos ^2\theta , \sin ^2\theta \}}\). By a similar argument, we take \(-\theta \in (-\frac{\pi }{2},0)\), and under the same condition on r, then \(D_{-\theta ,r}\) is contained in\(\{x\in {\mathbb {S}}^{N-1}; \ x_1<0, \ x_N>0\}\).
An immediate consequence of the previous remark and Lemma A.1 is the following:
Corollary A.3
Let \(\theta \in (\arcsin (\frac{1}{\sqrt{N}}), \frac{\pi }{2})\) and set \(r_\theta :=\sqrt{\max \{\cos ^2\theta , \sin ^2\theta \}}\). Then, for any \(r\in (r_\theta ,1)\), the spherical cap \(D_{\theta ,r}\) is contained in \(\{x\in {\mathbb {S}}^{N-1}; \ x_1>0, \ x_N>0\}\), while the symmetrical domain (with respect to the hyperplane \(\{x_1=0\}\)), namely \(D_{-\theta ,r}\), is contained in \(\{x\in {\mathbb {S}}^{N-1}; \ x_1<0, \ x_N>0\}\). Moreover, it holds that
and
Notice that (A.15) follows from the symmetry of \(D_{\theta ,r} \cup D_{-\theta ,r}\), as \(u_{{e}_1}\) is odd. Since \(D_{\theta ,r}\cup D_{-\theta ,r}\) is not connected, in order to apply the instability criterion given by Proposition 5.2 our idea is to join the two domains \(D_{\theta ,r}\), \(D_{-\theta ,r}\) by a suitably “small” tunnel-like domain which is symmetric with respect to the hyperplane \(\{x_1=0\}\). More precisely, we have the following:
Example A.4
Let \(\varepsilon >0\) be a small number to be determined later and consider the open region \(A_\varepsilon \subset {\mathbb {S}}^{N-1}\) between the two symmetric hyperplanes \(\{x_{N-1}=-\varepsilon \}\) and \(\{x_{N-1}=+\varepsilon \}\), namely
Setting \(\partial A_\varepsilon ^+:=\{x\in {\mathbb {S}}^{N-1}; \ x \varvec{\cdot }{e_{N-1}} =\varepsilon \}\), \(\partial A_\varepsilon ^-:=\{x\in {\mathbb {S}}^{N-1}; \ x \varvec{\cdot }(-{e_{N-1}}) =\varepsilon \}\) and arguing as in the proof of Lemma A.1 (see (A.4)) we can check that the exterior unit co-normal to \(\partial A_{\varepsilon }=\partial A_{\varepsilon }^-\cup A_\varepsilon ^+\), pointing outwards \(A_{\varepsilon }\), is given by
In view of (A.16), and as \({e}_1 \varvec{\cdot } {e}_{N-1}=0\), it follows that \(\nu (x) \varvec{\cdot } {e}_1= - \varepsilon x_1\) for all \(x\in \partial A_{\varepsilon }=\partial A_{\varepsilon }^-\cup A_\varepsilon ^+\), and thus
Now, fixing \(\theta \) and \(r\in (r_\theta ,1)\) as in Corollary A.3 we can choose \(\varepsilon >0\) sufficiently small (depending on r and \(\theta \)) so that \(\partial A_{\varepsilon }\) intersects \(D_{\theta ,r}\cup D_{-\theta ,r}\). We then take as tunnel-like domain the connected subset of
containing \({e}_N\), and we denote it by \(T_{\varepsilon ,\theta ,r}\). We set
By definition it is easy to check that \(D_{\varepsilon ,\theta ,r}\) is a domain symmetric with respect to the hyperplane \(\{x_1=0\}\) and thus, as \(u_{{e}_1}\) is odd, we have
Then by (A.14) and (A.17) we easily check that
if \(\varepsilon >0\) is sufficiently small, so that \(D_{\varepsilon , \theta ,r}\) satisfies (i) of Proposition 5.2 and hence \(\uplambda _1(D_{\varepsilon , \theta ,r})<N-1\). Moreover, by construction \(D_{\varepsilon , \theta ,r} \subset {\mathbb {S}}_+^{N-1}\) and also the inequality \({\mathcal {H}}_{N-1}(D_{\varepsilon , \theta ,r})<{\mathcal {H}}_{N-1}({\mathbb {S}}_+^{N-1})\) holds. Note that the domain \(D_{\varepsilon , \theta ,r}\) is not smooth but we can take a smooth domain close to \(D_{\varepsilon , \theta ,r}\) for which the same properties hold.
Next we exhibit another class of non-convex domain satisfying condition (i) of Proposition 5.2 which is not contained in a hemisphere.
Example A.5
Let us fix \(k\in \{1,\ldots ,N-2\}\) and let
Moreover, we fix \(r\in (0,\frac{\pi }{2})\) and consider
where \(\mathrm {dist}_{{\mathbb {S}}^{N-1}}\) denotes the geodesic distance in \({\mathbb {S}}^{N-1}\). If \(e\in {\mathbb {S}}^k\), we have \(\int _{D_r} u_e \ \mathrm{d}\sigma =0\) since \(D_r\) is symmetric with respect to reflection at the hyperplane
and \(u_e\) is odd with respect to this reflection. We write points in \({\mathbb {S}}^{N-1}\) as
with \(y\in {\mathbb {S}}^k\) (see (A.18)), \(\theta =\mathrm {dist}_{{\mathbb {S}}^{N-1}}(x,{\mathbb {S}}^k)\in (0,\frac{\pi }{2})\) and
In these coordinates, and since \(e\in {\mathbb {S}}^k\), we have \(u_e(x)= (e \varvec{\cdot } y)\cos \theta .\) In addition we check that
and the exterior unit co-normal in a point \(x=y\cos r + z \sin r \in \partial D_r\) is given by \(\nu (x)=-y \sin r + z\cos r\). Consequently, for any \(x\in \partial D_r\) we have
Hence it follows that
and thus
By Proposition 5.2, the inequality (A.19) implies that \(\uplambda _1(D_r)<N-1\). Finally if r is small, the condition \({\mathcal {H}}_{N-1}(D_r)<{\mathcal {H}}_{N-1}({\mathbb {S}}_+^{N-1})\) also holds.
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Iacopetti, A., Pacella, F. & Weth, T. Existence of Nonradial Domains for Overdetermined and Isoperimetric Problems in Nonconvex Cones. Arch Rational Mech Anal 245, 1005–1058 (2022). https://doi.org/10.1007/s00205-022-01801-4
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DOI: https://doi.org/10.1007/s00205-022-01801-4