1,721,016 research outputs found
Local behavior of fractional p-minimizers
Full text linkWe extend the De Giorgi–Nash–Moser theory to nonlocal, possibly degenerate integro-differential operators
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations
No full textWe deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order (Formula presented.) and summability growth (Formula presented.), 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
Holder continuity up to the boundary for a class of fractional obstacle problems
No full textWe deal with the obstacle problem for a class of nonlinear integro-differential 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 Holder continuity, up to the boundary. Key words: Quasilinear nonlocal operators, fractional Sobolev spaces, nonlocal tail, Caccioppoli estimates, obstacle problem
Nonlinear Potential Theory of elliptic systems
No full textWe give a summary of recent results from Nonlinear Potential Theory, focusing on linear and nonlinear potential estimates of solutions to non-homogeneous equations and systems. We start with the cases of quasilinear, possibly degenerate equations of p-Laplacian type, and, passing through fully nonlinear elliptic equations, finally move to systems. In this last case we describe recent results implying potential estimates for the p-Laplacian system. Finally, we describe results bridging Nonlinear Potential Theory and classical partial regularity theory for general elliptic systems. The main outcomes are new Ïμ-regularity criteria involving both excess functionals and nonlinear potentials
Borderline gradient continuity for nonlinear parabolic systems
No full textWe consider the evolutionary (Formula Presented)-Laplacean system (Formula Presented), and prove the continuity of the spatial gradient (Formula Presented) under the Lorentz space assumption (Formula Presented) is time independent the condition improves in (Formula Presented). This is the limiting case of a result of DiBenedetto claiming that (Formula Presented) is Hölder continuous when (Formula Presented) for (Formula Presented). At the same time, this is the natural nonlinear parabolic analog of a linear result of Stein, claiming the gradient continuity of solutions to the linear elliptic system (Formula Presented) is continuous. New potential estimates are derived and moreover suitable nonlinear potentials are used to describe fine properties of solutions
The obstacle problem for nonlinear integro-differential operators
Full text linkWe 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
Borderline gradient continuity of minima
No full textThe gradient of any local minimiser of functionals of the type (Formula presented.), where f has p-growth, p > 1, and (Formula presented.), is continuous provided that the optimal Lorentz space condition (Formula presented.) is satisfied and (Formula presented.) is suitably Dini continuous
Nonlinear Calderón-Zygmund Theory in the Limiting Case
No full textWe prove a maximal differentiability and regularity result for solutions to nonlinear measure data problems. Specifically, we deal with the limiting case of the classical theory of CalderÃ3n and Zygmund in the setting of nonlinear, possibly degenerate equations and we show a complete linearization effect with respect to the differentiability of solutions. A prototype of the results obtained here tells for instance that (Formula presented.) being a Borel measure with locally finite mass on the open subset Ω â Rnand p> 2 - 1 / n, then (Formula presented.) for every Ï â (0,1).The case Ï= 1 is obviously forbidden already in the classical linear case of the Poisson equation - âμu= Î1⁄4
A note on fractional supersolutions
Full text linkWe study a class of equations driven by nonlocal, possibly degenerate,
integro-differential operators of differentiability order
and summability growth , whose model is the fractional -Laplacian
with measurable coefficients. We prove that the minimum of the corresponding
weak supersolutions is a weak supersolution as well
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