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Isoperimetric inequalities for the fractional composite membrane problem

Mrityunjoy Ghosh

TL;DR

The article addresses isoperimetric-type questions for the first eigenvalue of the fractional composite membrane problem, defined via the fractional Laplacian with a Dirichlet condition and a region $D$ of fixed measure inside a domain $\Omega$. It proves a fractional Faber–Krahn inequality, showing the ball minimizes the first eigenvalue $\Lambda_{\Omega}(\alpha,c)$ among domains of fixed volume for any $\alpha>0$, with equality characterizing balls up to translations through the equality case of fractional Polya–Szegő. It also establishes a Lieb-type inequality for the intersection of two domains, demonstrating a strict bound on $\Lambda_{\Omega_1\cap\Omega_{2,x}}(\alpha,c)$ when translating one domain, using a nonlocal product-function construction and careful variational estimates; extensions to mixed local–nonlocal operators are discussed. The results extend classical isoperimetric inequalities to the fractional setting and illuminate how symmetrization and nonlocality interact in spectral optimization problems, offering new insights into parameter-independence and equality cases in fractional diffusion contexts.

Abstract

In this article, we investigate some isoperimetric-type inequalities related to the first eigenvalue of the fractional composite membrane problem. First, we establish an analogue of the renowned Faber-Krahn inequality for the fractional composite membrane problem. Next, we investigate an isoperimetric inequality for the first eigenvalue of the fractional composite membrane problem on the intersection of two domains-a problem that was first studied by Lieb [23] for the Laplacian. Similar results in the local case were previously obtained by Cupini-Vecchi [9] for the composite membrane problem. Our findings provide further insights into the fractional setting, offering a new perspective on these classical inequalities.

Isoperimetric inequalities for the fractional composite membrane problem

TL;DR

The article addresses isoperimetric-type questions for the first eigenvalue of the fractional composite membrane problem, defined via the fractional Laplacian with a Dirichlet condition and a region of fixed measure inside a domain . It proves a fractional Faber–Krahn inequality, showing the ball minimizes the first eigenvalue among domains of fixed volume for any , with equality characterizing balls up to translations through the equality case of fractional Polya–Szegő. It also establishes a Lieb-type inequality for the intersection of two domains, demonstrating a strict bound on when translating one domain, using a nonlocal product-function construction and careful variational estimates; extensions to mixed local–nonlocal operators are discussed. The results extend classical isoperimetric inequalities to the fractional setting and illuminate how symmetrization and nonlocality interact in spectral optimization problems, offering new insights into parameter-independence and equality cases in fractional diffusion contexts.

Abstract

In this article, we investigate some isoperimetric-type inequalities related to the first eigenvalue of the fractional composite membrane problem. First, we establish an analogue of the renowned Faber-Krahn inequality for the fractional composite membrane problem. Next, we investigate an isoperimetric inequality for the first eigenvalue of the fractional composite membrane problem on the intersection of two domains-a problem that was first studied by Lieb [23] for the Laplacian. Similar results in the local case were previously obtained by Cupini-Vecchi [9] for the composite membrane problem. Our findings provide further insights into the fractional setting, offering a new perspective on these classical inequalities.
Paper Structure (11 sections, 6 theorems, 54 equations)

This paper contains 11 sections, 6 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Faber-Krahn inequality
  3. Lieb inequality
  4. Preliminaries
  5. Fractional Sobolev spaces
  6. Symmetrization
  7. Properties of the minimizers
  8. Proofs of the Main Results
  9. Proof of Faber-Krahn inequality
  10. Proof of Lieb inequality
  11. Acknowledgements

Key Result

Theorem 1.1

Let $\Omega\subset {\mathbb R}^N$ be a bounded, Lipschitz domain and $\Omega^*$ be the ball centered at the origin satisfying $|\Omega|=|\Omega^*|$. Suppose $c\in (0,|\Omega^*|)$ and $\alpha>0$ are given. Then, we have the following where $\Lambda_{\Omega}(\alpha,c)$ is defined by Lam_om. Furthermore, equality occurs in FaberKrahn_ineq if and only if $\Omega=\Omega^*$ (up to translation).

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Rearrangement
  • Definition 2.2: Schwarz symmetrization
  • Lemma 2.3: Hardy-Littlewood inequality
  • proof
  • Lemma 2.4: Polya-Szeg̈o inequality
  • proof
  • Proposition 2.5
  • proof
  • ...and 6 more