We study the local regularity properties of (s, p)-harmonic functions, i.e. local weak solutions to the fractional p-Laplace equation of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$s\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in the case <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p\in (1,2]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. It is shown that (s, p)-harmonic functions are weakly differentiable and that the weak gradient is locally integrable to any power <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$q\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. As a result, (s, p)-harmonic functions are Hölder continuous to arbitrary Hölder 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$W^{2,2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>W</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-estimate.

Gradient regularity for (s, p)-harmonic functions

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发表日期：

2025年9月16日

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Verena Bögelein,
Frank Duzaar,
Naian Liao,
Giovanni Molica Bisci,
Raffaella Servadei

摘要

We study the local regularity properties of (s, p)-harmonic functions, i.e. local weak solutions to the fractional p-Laplace equation of order $s\in (0,1)$ $s \in (0, 1)$ in the case $p\in (1,2]$ $p \in (1, 2]$ . It is shown that (s, p)-harmonic functions are weakly differentiable and that the weak gradient is locally integrable to any power $q\ge 1$ $q \geq 1$ . As a result, (s, p)-harmonic functions are Hölder continuous to arbitrary Hölder 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 $W^{2,2}$ $W^{2, 2}$ -estimate.