Mixed local and nonlocal elliptic operators: regularity and maximum principles

Abstract

We develop a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.

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

“…This fact shows that, when Ω is sufficiently regular, X 1,p 0 (Ω) coincides with the space X p (Ω) introduced in [6] (for p = 2) and in [9] (for a general p > 1).…”

“…The case of mixed operators. The study of mixed local/nonlocal operators has been recently received an increasing level of attention, both in view of their intriguing mathematical structure, which combines the classical setting and the features typical of nonlocal operators in a framework that is not scale-invariant [40,45,46,5,32,10,21,4,20,24,23,22,39,7,1,18,30,27,28,35,36,37,38,19,9,6,54], and of their importance in practical applications such as the animal foraging hypothesis [29,51].…”

A Faber-Krahn inequality for mixed local and nonlocal operators

“…Then, by applying the Lax-Milgram Theorem to the bilinear form B Ω,s , one can prove the following existence result (see, e.g., [7,Theorem. 1.1]).…”

On the regularity theory for mixed local and nonlocal quasilinear parabolic equations

“…Existence and symmetry results together with various other qualitative properties of solutions of (1. 16) have recently been studied by Biagi-Dipierro-Valdinoci-Vecchi [6,7], Dipierro-Proietti Lippi-Valdinoci [22,23] and Dipierro-Ros-Oton-Serra-Valdinoci [24]. Much less is known in the nonlinear case p = 2 of (1.16).…”