Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators

Abstract

The purpose of this paper is to derive some Lewy–Stampacchia estimates in some cases of interest, such as the ones driven by non-local operators. Since we will perform an abstract approach to the problem, this will provide, as a byproduct, Lewy–Stampacchia estimates in more classical cases as well. In particular, we can recover the known estimates for the standard Laplacian, the $p$-Laplacian, and the Laplacian in the Heisenberg group. In the non-local framework we prove a Lewy–Stampacchia estimate for a general integrodifferential operator and, as a particular case, for the fractional Laplacian. As far as we know, the abstract framework and the results in the non-local setting are new.