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Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators. (English) Zbl 1275.49016
The paper under review establishes Lewy-Stampacchia type estimates for several classes of nonlinear problems, starting with classes driven by nonlocal operators. The approach developed in this paper is of high interest and it covers the settings corresponding to the standard Laplace or $p$-Laplace operators, as well as the Laplacian on the Heisenberg group. The case of integral operators with even kernel is also covered by the Lewy-Stampacchia estimates established in this paper. The proofs combine modern techniques in the theory of nonlocal operators with arguments in the theory of nonlinear partial differential equations.
##### MSC:
 49J40 Variational methods including variational inequalities 35J86 Linear elliptic unilateral problems; linear elliptic variational inequalities 35J87 Nonlinear elliptic unilateral problems; nonlinear elliptic variational inequalities 35R11 Fractional partial differential equations 35R35 Free boundary problems for PDE