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Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations. (English) Zbl 1386.35028

This paper deals with a class of nonlocal nonlinear integro-differential equations which involve fractional Laplacian-type operators with measurable coefficients. The nonlocal Dirichlet boundary value problems are defined and examined in the light of Perron method from nonlinear potential theory. The fractional operators oblige to take the nonlocal nature of the problem into account and give the corresponding definitions. The concept of \((s,p)\)-superharmonic and \((s,p)\)-subharmonic functions and accordingly nonlocal Perron method are introduced. The authors argue that, by the conclusions given, there is a remarkable contribution to the fractional nonlinear potential theory and since they work with a general class of nonlinear integro-differential operators with measurable coefficients, they provide alternative proofs for the particular cases appearing in the literature.

MSC:

35B45 A priori estimates in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35R05 PDEs with low regular coefficients and/or low regular data
47G20 Integro-differential operators
60J75 Jump processes (MSC2010)
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