Boundary regularity of mixed local-nonlocal operators and its application

Abstract

Let Ω be a bounded C 2 domain in R n and uin viscosity sense, where 0 ≤ a ≤ A0, C0, K > 0, and I is a suitable nonlocal operator. We show that u/δ is in C κ ( Ω) for some κ ∈ (0, 1), where δ(x) = dist(x, Ω c ). Using this result we also establish that u ∈ C 1,γ ( Ω). Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators.

Necessary condition in a Brezis–Oswald-type problem for mixed local and nonlocal operators

“…(2) As pointed out in [4], assumption (f5) and the two-side growth condition in assumption (f3) are technical assumption which permit to overcome the lack of boundary regularity for L p,s , which is instead a crucial tool in [7,11,12]. Presently, the regularity for L p,s is deeply investigated, see [1,2,3,8,10,9,13,14] for the case of weak solutions and [5] for the case of viscosity solutions; however, the optimal boundary regularity for L p,s in the context of weak solutions and a Hopf-type lemma seem lacking. As it will be clear from the proof of Theorem 1.3, assumptions (f3)-(f5) allows us to set up a suitable truncation/approximation argument which turns out to be a proper substitute of a Hopf-type lemma for L p,s .…”

Variational methods for nonpositive mixed local-nonlocal operators

“…At the present stage, and without aim of completeness, the investigations have taken into consideration interior regularity and maximum principles (see e.g. [2,9,13,17,24,25]), boundary Harnack principle [14], boundary regularity and overdetermined problems [10,36], qualitative properties of solutions [3], existence of solutions and asymptotics (see e.g. [6,7,8,12,15,20,21,26,33,35]) and shape optimization problems [4,5,27].…”