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Gradient regularity for (s, p)-harmonic functions
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DOI：10.1007/s00526-025-03116-0.png)
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Abstract
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We study the local regularity properties of (s, p)-harmonic functions, i.e. local weak solutions to the fractional p-Laplace equation of order s is an element of(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} in the case p is an element of(1,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (1,2]$$\end{document}. It is shown that (s, p)-harmonic functions are weakly differentiable and that the weak gradient is locally integrable to any power q >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\ge 1$$\end{document}. As a result, (s, p)-harmonic functions are H & ouml;lder continuous to arbitrary H & ouml;lder exponent in (0, 1). In addition, the weak gradient of (s, p)-harmonic functions has certain fractional differentiability. All estimates are stable when s reaches 1, and the known regularity properties of p-harmonic functions are formally recovered, in particular the local W2,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W<^>{2,2}$$\end{document}-estimate.
Keywords:
FINE BOUNDARY-REGULARITY
FRACTIONAL LAPLACIAN
NONLOCAL EQUATIONS
HOLDER REGULARITY
MU-TRANSMISSION
P-LAPLACIAN
MINIMIZERS
SOBOLEV
Journal
C
IF:
2
Papers:
247
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0
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