Nontrivial solutions to Serrin's problem in annular domains

doi:10.1016/j.anihpc.2020.05.001

Annales de l'Institut Henri Poincaré C, Analyse non linéaire

Volume 38, Issue 1, January–February 2021, Pages 1-22

https://doi.org/10.1016/j.anihpc.2020.05.001 Get rights and content

Abstract

We construct nontrivial bounded, real analytic domains $Ω \subseteq R^{n}$ of the form $Ω_{0} ∖ {\overline{Ω}}_{1}$ , bifurcating from annuli, which admit a positive solution to the overdetermined boundary value problem ${\begin{matrix} - Δ u = 1, u > 0 & in Ω, \\ u = 0, \partial_{ν} u = const & on \partial Ω_{0}, \\ u = const, \partial_{ν} u = const & on \partial Ω_{1}, \end{matrix}$ where ν stands for the inner unit normal to ∂Ω. From results by Reichel [1] and later by Sirakov [2], it was known that the condition $\partial_{ν} u \leq 0$ on $\partial Ω_{1}$ is sufficient for rigidity to hold, namely, the only domains which admit such a solution are annuli and solutions are radially symmetric. Our construction shows that the condition is also necessary. In addition, we show that the constructed domains are self-Cheeger.

Section snippets

Introduction and main result

Let $Ω \subseteq R^{n}$ be a bounded, connected $C^{2}$ -domain of the form $Ω = Ω_{0} ∖ {\overline{Ω}}_{1}$ , where $Ω_{0}$ and $Ω_{1}$ are bounded domains in $R^{n}$ , $n \geq 2$ , with $Ω_{1} ⋐ Ω_{0}$ . The present paper is devoted to the overtermined boundary value problem ${\begin{matrix} - Δ u = 1 & in Ω, \\ u = 0, \partial_{ν} u = c_{0} & on \partial Ω_{0}, \\ u = a, \partial_{ν} u = c_{1} & on \partial Ω_{1}, \end{matrix}$ where ν denotes the inner unit normal to Ω and $a \geq 0$ , $c_{0}$ and $c_{1}$ are real constants. Note that whenever (1.1) admits a solution $u \in C^{2} (\overline{Ω})$ , then u is strictly positive in Ω due the strong maximum principle, while the Hopf lemma implies that the constant $c_{0} > 0$ .

Acknowledgements

The authors would like to thank Dimiter Vassilev for bringing up Serrin's problem in annular domains to their attention. We are grateful to Martin Chuaqui for several useful discussions. We would also like to thank the referees for helpful comments leading to an improved presentation.

Outline of strategy and refinement of the main theorem

Let us first introduce some notation. For any $λ \in (0, 1)$ we denote the standard annulus of inner radius λ and outer radius 1 by $Ω_{λ} : = {x \in R^{n} : λ < | x | < 1}$ and let its two boundary components be $Γ_{1} : = {x \in R^{n} : | x | = 1} = S^{n - 1}, Γ_{λ} : = {x \in R^{n} : | x | = λ} = λ S^{n - 1},$ where $S^{n - 1}$ is the unit sphere in $R^{n}$ , centered at the origin.

We will construct the nontrivial solutions u and domains Ω solving (1.1)-(1.2) by bifurcating away, at certain critical values of the bifurcation parameter λ, from the branch of non-monotone radial solutions $u_{λ}$

Reformulating the problem and deriving its linearization

Let us first recast the operator $F_{λ}$ , defined in (2.6), by pulling back the Dirichlet problem (2.5) from $Ω_{λ}^{v}$ to the annulus $Ω_{λ}$ , where we shall use polar coordinates $(0, \infty) \times S^{n - 1} ≅ R^{n} ∖ {0} under (r, θ) \mapsto x = r θ$ to describe the geometry. In this way, $Ω_{λ} ≅ (λ, 1) \times S^{n - 1}$ , its boundary components $Γ_{λ} ≅ {λ} \times S^{n - 1}$ , $Γ_{1} ≅ {1} \times S^{n - 1}$ are two copies of $S^{n - 1}$ , and we naturally get the identification of functions (2.9).

For any $v = (v_{1}, v_{2}) \in U \subseteq {(C^{2, α} (S^{n - 1}))}^{2}$ of sufficiently small norm, we consider the diffeomorphism $Φ : Ω_{λ} \to Ω_{λ}^{v}$ defined in

Spectrum of the linearized operator

In this section we give an account of the spectral properties of the linearized operator $L_{λ}$ , which we derived in Proposition 3.1.

Recall that a function $Y \in C^{\infty} (S^{n - 1})$ is a spherical harmonic of degree $k \in N_{0}$ if it is an eigenfunction of the Laplace-Beltrami operator $- Δ_{S^{n - 1}}$ on $S^{n - 1}$ , that is, $Δ_{S^{n - 1}} Y + σ_{k} Y = 0,$ where $σ_{k} : = k (k + n - 2)$ is the corresponding eigenvalue. We first observe that the subspace W generated by ${(Y, 0), (0, Y)}$ is invariant under $L_{λ}$ and we shall derive a matrix representation of $L_{λ} |_{W}$ with

The proof of Theorem 2.3

We now turn to the proof of Theorem 2.3. Following the discussion given in Section 2, it will be necessary to specialize to functions that are invariant under the action of a subgroup G of the orthogonal group $O (n)$ satisfying (P1)-(P2), stated in Section 2. Recall that $C_{G}^{k, α} (S^{n - 1})$ denotes the Hölder space of G-invariant functions.

We begin by observing that the operator $F_{λ}$ defined in (3.5) restricts to the G-invariant function spaces ${(C_{G}^{k, α} (S^{n - 1}))}^{2}$ and, therefore, so does its linearization $L_{λ}$ .

Lemma 5.1

Proof of Corollary 1.2

Let Ω be any one of the domains constructed in Theorem 1.1 and let $u \in C^{\infty} (\overline{Ω})$ be the solution of the corresponding overdetermined problem ${\begin{matrix} - Δ u = 1 & in Ω, \\ u = 0 & on \partial Ω_{0}, \\ u = a & on \partial Ω_{1}, \\ \partial_{ν} u = c & on \partial Ω, \end{matrix}$ for some constants $a > 0$ and $c > 0$ . The corollary now follows directly from the more general [33, Theorem 1.2]. For the sake of completeness, we shall provide the proof in our particular setting.

Proof of Corollary 1.2

First, let us show that $| \nabla u | < c in Ω .$ Indeed, since $- Δ u = 1$ , $Δ | \nabla u |^{2} = 2 | D^{2} u |^{2} + 2 \nabla u \cdot \nabla (Δ u) = 2 | D^{2} u |^{2} > 0,$ so that the function $| \nabla u |^{2}$ is

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

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¹: The first author acknowledges the support of Proyecto FONDECYT Iniciación No. 11160981.

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Nontrivial solutions to Serrin's problem in annular domains

Abstract

Section snippets

Introduction and main result

Acknowledgements

Outline of strategy and refinement of the main theorem

Reformulating the problem and deriving its linearization

Spectrum of the linearized operator

The proof of Theorem 2.3

Proof of Corollary 1.2

Declaration of Competing Interest

Ann. Inst. Henri Poincaré, Anal. Non Linéaire

J. Funct. Anal.

Adv. Math.

Ann. Inst. Henri Poincaré, Anal. Non Linéaire

Radial symmetry by moving planes for semilinear elliptic BVPs on annuli and other non-convex domains

Overdetermined elliptic problems in physics

A symmetry problem in potential theory

Arch. Ration. Mech. Anal.

A characteristic property of spheres

Ann. Mat. Pura Appl.

Symmetry theorems for some overdetermined boundary value problems on ring domains

Z. Angew. Math. Phys.

Overdetermined boundary value problems, quadrature domains and applications

Comput. Methods Funct. Theory