On a Rayleigh-Faber-Krahn inequality for the regional fractional Laplacian

TL;DR

This paper investigates a Rayleigh-Faber-Krahn inequality for regional fractional Laplacians, establishing the existence of minimizers with specific properties using direct methods, and discusses open questions about their regularity and shape.

Contribution

It introduces a new approach based on direct methods and diameter estimates for regional fractional Laplacians, differing from traditional symmetrization techniques.

Findings

Existence of a compactly supported minimizer for the regional fractional Laplacian problem.

Development of a new approach using a priori diameter estimates.

Open questions on regularity, shape, and Euler-Lagrange equations of minimizers.

Abstract

We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian operators. In particular, we show that there exists a compactly supported nonnegative Sobolev function $u_0$ that attains the infimum (which will be a positive real number) of the set \[ \left\{ \int\int_{\{u > 0\}\times\{u>0\}} \frac{|u(x) - u(y)|^2}{|x - y|^{n + 2 \sigma}}d x d y : u \in \mathring H^\sigma(\mathbb{R}^n), \int_{\mathbb{R}^n} u^2 = 1, |\{u > 0 \}| \leq 1\right\}. \] Unlike the corresponding problem for the usual fractional Laplacian, where the domain of the integration is $\mathbb{R}^n \times \mathbb{R}^n$, symmetrization techniques may not apply here. Our approach is instead based on the direct method and new a priori diameter estimates. We also present several remaining open questions concerning the regularity and shape of the minimizers, and the form of the Euler-Lagrange equations.