<p>This article features a case study of the following four relevant subjects: (i) the hypersurface potential <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40879_2025_867_IEq1_HTML.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="120" Type="Linedraw" Width="35" /> </InlineMediaObject> </InlineEquation> on an <i>m</i>-homogeneous graph <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40879_2025_867_IEq2_HTML.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="120" Type="Linedraw" Width="204" /> </InlineMediaObject> </InlineEquation>; (ii) the hypersurface capacity <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40879_2025_867_IEq3_HTML.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="120" Type="Linedraw" Width="31" /> </InlineMediaObject> </InlineEquation> on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>; (iii) the wave potential <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="MediaObjects/40879_2025_867_IEq5_HTML.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="120" Type="Linedraw" Width="21" /> </InlineMediaObject> </InlineEquation> over a light cone <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\{y\in \mathbb R^n\hspace{0.55542pt}{:}\, |y|<t\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>y</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> <mspace width="0.55542pt" /> <mo>:</mo> <mspace width="0.166667em" /> <mo stretchy="false">|</mo> <mi>y</mi> <mo stretchy="false">|</mo> <mo><</mo> <mi>t</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> under the spherical symmetry with respect to the space variable; (iv) the existence and uniqueness of a weak solution to the model semilinear wave equation <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Box \hspace{1.111pt}u=u|u|^{\kappa -1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>□</mo> <mspace width="1.111pt" /> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mi>u</mi> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>κ</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb R^3\hspace{0.55542pt}{\times }\hspace{1.111pt}(0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="double-struck">R</mi> <mn>3</mn> </msup> <mspace width="0.55542pt" /> <mo>×</mo> <mspace width="1.111pt" /> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with rough-to-rougher Sobolev data.</p>

Hypersurface potentials and capacities via semilinear wave equations

期刊论文
发表日期：

2025年10月17日

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David R. Adams,
Jie Xiao

摘要

This article features a case study of the following four relevant subjects: (i) the hypersurface potential on an m-homogeneous graph ; (ii) the hypersurface capacity on $\Sigma $ $Σ$ ; (iii) the wave potential over a light cone $\{y\in \mathbb R^n\hspace{0.55542pt}{:}\, |y|<t\}$ ${y \in R^{n} : | y | < t}$ under the spherical symmetry with respect to the space variable; (iv) the existence and uniqueness of a weak solution to the model semilinear wave equation $\Box \hspace{1.111pt}u=u|u|^{\kappa -1}$ $□ u = {u | u |}^{κ - 1}$ in $\mathbb R^3\hspace{0.55542pt}{\times }\hspace{1.111pt}(0,\infty )$ $R^{3} \times (0, \infty)$ with rough-to-rougher Sobolev data.