Qualche tempo dopo Stampacchia, partendo sempre dalla sua disequazione variazionale, aperse un nuovo campo di ricerche che si rivelò importante e fecondo. Si tratta di quello che oggiè chiamato "il problema dell'ostacolo".[ Sandro Faedo, 1987 ] Abstract We investigate the obstacle problem for a class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian operator with measurable coefficients. Amongst other results, we will prove both the existence and uniqueness of the solutions to the obstacle problem, and that these solutions inherit regularity properties, such as boundedness, continuity and Hölder continuity (up to the boundary), from the obstacle.
We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order s ∈ (0, 1) and summability growth p > 1, whose model is the fractional p-Laplacian with measurable coefficients. We state and prove several results for the corresponding weak supersolutions, as comparison principles, a priori bounds, lower semicontinuity, and many others. We then discuss the good definition of (s, p)-superharmonic functions, by also proving some related properties. We finally introduce the nonlocal counterpart of the celebrated Perron method in nonlinear Potential Theory.Keywords Quasilinear nonlocal operators · fractional Sobolev spaces · fractional Laplacian · nonlocal tail · Caccioppoli estimates · obstacle problem · comparison estimates · fractional superharmonic functions · the Perron Method
This paper studies smoothing properties of the local fractional maximal
operator, which is defined in a proper subdomain of the Euclidean space. We
prove new pointwise estimates for the weak gradient of the maximal function,
which imply norm estimates in Sobolev spaces. An unexpected feature is that
these estimates contain extra terms involving spherical and fractional maximal
functions. Moreover, we construct several explicit examples which show that our
results are essentially optimal. Extensions to metric measure spaces are also
discussed.Comment: 23 page
Abstract. In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional p-Laplace type P.V.Solutions are defined via integration by parts with test functions, as viscosity solutions or via comparison. Our main result states that for bounded solutions, the three different notions coincide.
We deal with the obstacle problem for a class of nonlinear integrodifferential operators, whose model is the fractional p-Laplacian with measurable coefficients. In accordance with well-known results for the analog for the pure fractional Laplacian operator, the corresponding solutions inherit regularity properties from the obstacle, both in the case of boundedness, continuity, and Hölder continuity, up to the boundary.
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