<p>This paper provides a deep understanding of the nonlinear potential theory for the fractional Hajłasz-Sobolev space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {W}^{\alpha ,p}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">W</mi> </mrow> <mrow> <mi>α</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on metric spaces using the fractional Sobolev capacity. To do so, the fractional Hajłasz-Sobolev space with zero boundary values <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {W}^{\alpha ,p}_{0}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="script">W</mi> </mrow> <mn>0</mn> <mrow> <mi>α</mi> <mo>,</mo> <mi>p</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is introduced and the compactness, removable sets, approximation by Lipschitz continuous functions, of this space are studied. As applications, the authors consider the fractional obstacle problem <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varPhi _{\psi ,h}(\varOmega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Φ</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>h</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {W}^{\alpha ,p}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">W</mi> </mrow> <mrow> <mi>α</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and prove the existence and uniqueness of solution for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varPhi _{\psi ,h}(\varOmega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Φ</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>h</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In particular, some special case of solutions for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varPhi _{\psi ,h}(\varOmega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Φ</mi> <mrow> <mi>ψ</mi> <mo>,</mo> <mi>h</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>(which are called fractional superminimizers) are investigated systematically by the fractional De Giorgi class, including the regularity, the lower semi continuous representation and some convergence results.</p>

Nonlinear potential theory for the fractional Hajłasz-Sobolev spaces on metric spaces

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发表日期：

2026年2月18日

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Shaoguang Shi,
Guoliang Li,
Wen Yuan

摘要

This paper provides a deep understanding of the nonlinear potential theory for the fractional Hajłasz-Sobolev space $\mathcal {W}^{\alpha ,p}(X)$ $W^{α, p} (X)$ on metric spaces using the fractional Sobolev capacity. To do so, the fractional Hajłasz-Sobolev space with zero boundary values $\mathcal {W}^{\alpha ,p}_{0}(X)$ $W_{0}^{α, p} (X)$ is introduced and the compactness, removable sets, approximation by Lipschitz continuous functions, of this space are studied. As applications, the authors consider the fractional obstacle problem $\varPhi _{\psi ,h}(\varOmega )$ $Φ_{ψ, h} (Ω)$ on $\mathcal {W}^{\alpha ,p}(X)$ $W^{α, p} (X)$ and prove the existence and uniqueness of solution for $\varPhi _{\psi ,h}(\varOmega )$ $Φ_{ψ, h} (Ω)$ . In particular, some special case of solutions for $\varPhi _{\psi ,h}(\varOmega )$ $Φ_{ψ, h} (Ω)$ (which are called fractional superminimizers) are investigated systematically by the fractional De Giorgi class, including the regularity, the lower semi continuous representation and some convergence results.