In this paper we consider the setting of a locally compact, non-complete metric measure space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(Z,d,\nu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>Z</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> equipped with a doubling measure <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>, under the condition that the boundary <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\partial Z:=\overline{Z}\setminus Z$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>Z</mi> <mo>:</mo> <mo>=</mo> <mover> <mi>Z</mi> <mo>¯</mo> </mover> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>Z</mi> </mrow> </math></EquationSource> </InlineEquation> (obtained by considering the completion of Z) supports a Radon measure <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation> which is in a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-codimensional relationship to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\sigma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We explore existence, uniqueness, comparison property, and stability properties of solutions to inhomogeneous Dirichlet problems associated with certain nonlinear nonlocal operators on Z. We also establish interior regularity of solutions when the inhomogeneity data is in an <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^q$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation>-class for sufficiently large <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$q>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and verify a Kellogg-type property when the inhomogeneity data vanishes and the Dirichlet data is continuous.

Well-posedness of Dirichlet boundary value problems for reflected fractional p-Laplace-type inhomogeneous equations in compact doubling metric measure spaces

期刊论文
发表日期：

2026年1月12日

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Josh Kline,
Feng Li,
Nageswari Shanmugalingam

摘要

In this paper we consider the setting of a locally compact, non-complete metric measure space $(Z,d,\nu )$ $(Z, d, ν)$ equipped with a doubling measure $\nu $ $ν$ , under the condition that the boundary $\partial Z:=\overline{Z}\setminus Z$ $\partial Z : = \bar{Z} \ Z$ (obtained by considering the completion of Z) supports a Radon measure $\pi $ $π$ which is in a $\sigma $ $σ$ -codimensional relationship to $\nu $ $ν$ for some $\sigma >0$ $σ > 0$ . We explore existence, uniqueness, comparison property, and stability properties of solutions to inhomogeneous Dirichlet problems associated with certain nonlinear nonlocal operators on Z. We also establish interior regularity of solutions when the inhomogeneity data is in an $L^q$ $L^{q}$ -class for sufficiently large $q>1$ $q > 1$ , and verify a Kellogg-type property when the inhomogeneity data vanishes and the Dirichlet data is continuous.