Hölder estimates for viscosity solutions of equations of fractional p-Laplace type

By Erik Lindgren. Nonlinear Differential Equations and Applications NoDEA, 2014

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We prove Hölder estimates for viscosity solutions of a class of possibly degenerate and singular equations modelled by the fractional p-Laplace equation PV∫Rn|u(x)-u(x+y)|p-2(u(x)-u(x+y))|y|n+spdy=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PV {\int_{\mathbb{R}^{n}}\frac{|u(x)-u(x + y)|^{p-2}(u(x) - u(x + y))}{|y|^{n+sp}}\,dy = 0},$$\end{document}where s∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s \in (0,1)}$$\end{document} and p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p > 2}$$\end{document} or 1/(1-s)<p<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1/(1-s) < p < 2}$$\end{document}. Our results also apply for inhomogeneous equations with more general kernels, when p and s are allowed to vary with x, without any regularity assumption on p and s. This complements and extends some of the recently obtained Hölder estimates for weak solutions.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/d933398643efd29cd0e0228da64a7fe74136e1bb

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Citations in the corpus

Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case (2017, background): “g. The same authors pursued their investigation in the ﬁeld, by proving Harnack’s inequality for solutions of the homogeneous equation in [17]. There are also many other recent regularity results. In [29], the second author studied the local Ho¨lder regularity, using viscosity methods. In [13], the ”
Hölder regularity for nonlocal double phase equations (2019, methodology): “ 0) u in Rn \ B′(x 0). Since u is a viscosity subsolution in the sense of Deﬁnition 1, then (3.1) Lϕ′(x 0) ≤ C. The proof will now follows that of the analogous result for the p-Laplace equation in [26], which extends to the nonlinear case the original proofby Caﬀarelli and Silvestre in [9, Lemma 3.”
Continuity of solutions to a nonlinear fractional diffusion equation (2019, background): “The regularity of solutions has been studied for instance in [3, 4, 6, 12, 13, 17, 16, 19, 20, 21, 24, 27] and [34].”
Hölder regularity for weak solutions to nonlocal double phase problems (2021, background): “We also refer to [20, 22, 27, 28, 33, 34, 35] and references therein for various regularity results for nonlocal problems of fractional p-Laplacian type.”
Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case (2015, background): “We start with the case where our solution u is a priori known to be globally bounded, a situation that is quite natural if u is constructed through viscosity methods (see [20]).”