A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

Abstract

<abstract><p>Given a bounded open set $ \Omega\subseteq{\mathbb{R}}^n $, we consider the eigenvalue problem for a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $ \Omega $. We prove that the second eigenvalue $ \lambda_2(\Omega) $ is always strictly larger than the first eigenvalue $ \lambda_1(B) $ of a ball $ B $ with volume half of that of $ \Omega $. This bound is proven to be sharp, by comparing to the limit case in which $ \Omega $ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.</p></abstract>

On the mixed local-nonlocal Hénon equation

“…Further regularity results for p = 2 are contained in [3], where the authors prove local H k+2 estimates and several maximum principles. Qualitative properties of solutions of semilinear equations in the spirit of the classical results by Gidas, Ni and Nirenberg have been proved in [4], while a quantitative version of a Faber-Krahn inequality has been proved in [5]. We refer also to [28] for related results.…”

Mixed local-nonlocal operators: maximum principles, eigenvalue problems and their applications

“…al. [7] establish Faber-Krahn inequality for the operator −∆ + (−∆) s for s ∈ (0, 1). Their method uses Schwarz symmetrization combined with the Polya-Szegö inequality and [30,Theorem A.1].…”

“…There is a large body of works dealing with elliptic operators with both local and nonlocal parts. But most of the works restricted the nonlocal term to be the factional Laplacian [1,3,4,7,9,22,25,43]. However, there are many practical situations; for instance, in biology [23,40], mathematical finance [14,41], where the Lévy measure need not be of fractional Laplacian type.…”

“…Their method uses Schwarz symmetrization combined with the Polya-Szegö inequality and [30,Theorem A.1]. Since the inequality in [30,Theorem A.1] holds for a more general class of kernel j, it might be possible to mimic the proof of [7] in an appropriate variational set-up and Sobolev space to establish (1.8). However, our viscosity solution approach does not impose any additional regularity on the solution.…”

A Brezis-Oswald approach for mixed local and nonlocal operators

“…Here, ∆ p u = div(|∇u| p−2 ∇u) is the classical p−Laplacian operator and, for fixed s ∈ (0, 1) and up to a multiplicative positive constant, the fractional p−Laplacian is defined as (−∆) s p u(x) := 2 P.V. Problems driven by operators like L p,s have raised a certain interest in the last few years, both for the mathematical complications that the combination of two so different operators imply and for the wide range of applications, see for instance [5,4,6,11,12,13,14] and the references therein. A common feature of the aforementioned papers is to deal with weak solutions, in contrast with other results existing in the literature where viscosity solutions have been considered, see e.g.…”