<p>In this paper, we investigate the boundary Hölder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the <i>p</i>-Laplace equations and fully nonlinear elliptic equations) on Reifenberg flat domains. We prove that for any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0<\alpha <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, there exists <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\delta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> such that the solution is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation> at <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$x_0\in \partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>∈</mo> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> provided that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-Reifenberg flat at <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$x_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>x</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> (see Definition <InternalRef RefID="FPar1">1.1</InternalRef>). Moreover, we have an explicit relation between <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. As a special case, for any <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$0< \alpha < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, if <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$C^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$u=g$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>=</mo> <mi>g</mi> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$g\in C^{\alpha }(x_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <msup> <mi>C</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$u\in C^{\alpha }(x_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <msup> <mi>C</mi> <mi>α</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Boundary Hölder regularity for elliptic equations on Reifenberg flat domains

期刊论文
发表日期：

2026年2月21日

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Yuanyuan Lian,
Kai Zhang

摘要

In this paper, we investigate the boundary Hölder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the p-Laplace equations and fully nonlinear elliptic equations) on Reifenberg flat domains. We prove that for any $0<\alpha <1$ $0 < α < 1$ , there exists $\delta >0$ $δ > 0$ such that the solution is $C^{\alpha }$ $C^{α}$ at $x_0\in \partial \Omega $ $x_{0} \in \partial Ω$ provided that $\Omega $ $Ω$ is $\delta $ $δ$ -Reifenberg flat at $x_0$ $x_{0}$ (see Definition 1.1). Moreover, we have an explicit relation between $\delta $ $δ$ and $\alpha $ $α$ . As a special case, for any $0< \alpha < 1$ $0 < α < 1$ , if $\partial \Omega $ $\partial Ω$ is $C^1$ $C^{1}$ and $u=g$ $u = g$ on $\partial \Omega $ $\partial Ω$ with $g\in C^{\alpha }(x_0)$ $g \in C^{α} (x_{0})$ , then $u\in C^{\alpha }(x_0)$ $u \in C^{α} (x_{0})$ .