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 -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.
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Raffaella Servadei, Enrico Valdinoci, Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators. Rev. Mat. Iberoam. 29 (2013), no. 3, pp. 1091–1126DOI 10.4171/RMI/750