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

By Stefano Biagi, Serena Dipierro, Enrico Valdinoci, E. Vecchi. Mathematics in Engineering, 2021

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3e6eda7d73291e5c48de5c6ac988e1f893942eb6

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Citations in the corpus

Regularity results for solutions of mixed local and nonlocal elliptic equations (2022, background): “Moreover, mixed operators have recently received a great attention from different points of view, including regularity theory [2, 3, 8, 10, 12, 14, 15, 17, 25], existence and non-existence results [1, 6, 28], eigenvalue problems [13, 16, 22], shape optimization and calculus of variations [7, 11], sy”
Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations (2022, background): “[5] has obtained local Hölder continuity for globally bounded solutions and Garain-Ukhlov [31] studied existence, uniqueness, local boundedness and further qualitative properties of solutions.”
On the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, background): “BiagiDipierro-Valdinoci-Vecchi [6] has obtained interior regularity results for a nonhomogeneous version of (1.”
Gradient regularity in mixed local and nonlocal problems (2022, background): “Compare for instance with the discussion made in [5,8], where related functional settings are considered.”