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Elliptic and parabolic overdetermined problems in multi-phase settings

Lorenzo Cavallina, Giorgio Poggesi

TL;DR

The paper establishes symmetry results for overdetermined elliptic and parabolic problems in multi-phase domains, including infinitely many phases and rough interfaces, by a direct weak-formulation approach. The main elliptic result shows that solvability forces rotational symmetry: either the domain is a ball or annulus, or the core comprises a single concentric ball with the shell, depending on whether the phase index set is empty or single. Parabolic counterparts are obtained by time-integrating the solution to reduce to the elliptic case, yielding the same symmetry dichotomy. These findings generalize classical Serrin-Sakaguchi results to complex multi-phase geometries and provide tools for diffusion-like problems in composite media.

Abstract

The present paper provides symmetry results for a class of overdetermined problems of elliptic and parabolic type in multi-phase settings, including various extensions of remarkable results obtained by S. Sakaguchi in [12, 13]. A new alternative approach to proving this type of results is presented, leveraging the weak formulation of the problem. The resulting proofs are direct and elegant, and bring several benefits, including extensions to multi-phase settings (possibly with infinitely many phases) and generalizations to rough interfaces.

Elliptic and parabolic overdetermined problems in multi-phase settings

TL;DR

The paper establishes symmetry results for overdetermined elliptic and parabolic problems in multi-phase domains, including infinitely many phases and rough interfaces, by a direct weak-formulation approach. The main elliptic result shows that solvability forces rotational symmetry: either the domain is a ball or annulus, or the core comprises a single concentric ball with the shell, depending on whether the phase index set is empty or single. Parabolic counterparts are obtained by time-integrating the solution to reduce to the elliptic case, yielding the same symmetry dichotomy. These findings generalize classical Serrin-Sakaguchi results to complex multi-phase geometries and provide tools for diffusion-like problems in composite media.

Abstract

The present paper provides symmetry results for a class of overdetermined problems of elliptic and parabolic type in multi-phase settings, including various extensions of remarkable results obtained by S. Sakaguchi in [12, 13]. A new alternative approach to proving this type of results is presented, leveraging the weak formulation of the problem. The resulting proofs are direct and elegant, and bring several benefits, including extensions to multi-phase settings (possibly with infinitely many phases) and generalizations to rough interfaces.

Paper Structure

This paper contains 8 sections, 11 theorems, 83 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main result concerning an elliptic overdetermined problem
  3. Applications to the parabolic setting
  4. Organization of the paper
  5. Some preliminary results
  6. Proof of Theorem \ref{['mainthm I']}
  7. Some applications to parabolic overdetermined problems
  8. Integrating the solution of the multi-phase heat equation

Key Result

Theorem A

Let $\Omega$ be a ball and let $\emptyset \neq D \subset \subset \Omega$ be the disjoint union of finitely many domains of class $C^2$. Assume that $\Omega \setminus D$ is connected. Then, the overdetermined problem eq:2 phase pb--eq:Serrin over admits a solution if and only if $(D,\Omega)$ are conc

Figures (2)

  • Figure 1: The geometrical interpretation of the discreteness assumption \ref{['discreteness']}. Here examples $(i)$ and $(ii)$ satisfy \ref{['discreteness']}, while examples $(iii)$ and $(iv)$ do not. Notice that in $(iii)$, two distinct phases of the core touch, thus violating \ref{['discreteness']}. On the other hand, $(iv)$ still violates \ref{['discreteness']} without two distinct phases of the core touching.
  • Figure 2: The geometric setting of our problem. Notice that we do not impose any regularity assumption on $\Gamma$, allowing even fractal interfaces.

Theorems & Definitions (26)

  • Theorem A: sakaguchibessatsu
  • Remark 1.1
  • Theorem I
  • Remark 1.2
  • Theorem II
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 16 more