Revista Matemática Iberoamericana

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Volume 29, Issue 3, 2013, pp. 1091–1126
DOI: 10.4171/RMI/750

Published online: 2013-08-04

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

Raffaella Servadei[1] and Enrico Valdinoci[2]

(1) Università della Calabria, Cosenza, Italy
(2) Università degli Studi di Milano, Italy

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.

Keywords: Variational inequalities, integrodifferential operators, fractional Laplacian

Servadei Raffaella, Valdinoci Enrico: Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators. Rev. Mat. Iberoamericana 29 (2013), 1091-1126. doi: 10.4171/RMI/750