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

Abstract

Given a bounded open set Ω ⊆ R n , we consider the eigenvalue problem of a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of Ω.We prove that the second eigenvalue λ2(Ω) is always strictly larger than the first eigenvalue λ1(B) of a ball B with volume half of that of Ω.This bound is proven to be sharp, by comparing to the limit case in which Ω 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.

Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations

“…In [7], Biagi-Mugnai-Vecchi established boundedness and strong maximum principle in the inhomogeneous case. In the case of a bounded function f , Biagi-Dipierro-Valdinoci-Vecchi [5] has obtained local Hölder continuity for globally bounded solutions and Garain-Ukhlov [30] studied existence, uniqueness, local boundedness and further qualitative properties of solutions. Moreover, for more general inhomogeneites, local boundedness is proved in Salort-Vecchi [33].…”