In this paper, we consider an extremal problem associated to the solution of a boundary value problem on a convex domain <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. Our main focus is on establishing a variational formula for a functional related to the p-harmonic measure of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, from which a new surface-area-type measure of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is derived. This further motivates us to study the Minkowski problem for this new measure. As a main result, we prove the existence of solutions to the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> Minkowski problem associated to the p-harmonic measure for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$0<q<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$1<p\ne n+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo>≠</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.

The $L_q$ Minkowski problem for p-harmonic measure

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发表日期：

2026年3月14日

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Hai Li,
Longyu Wu,
Baocheng Zhu

摘要

In this paper, we consider an extremal problem associated to the solution of a boundary value problem on a convex domain $\Omega $ $Ω$ . Our main focus is on establishing a variational formula for a functional related to the p-harmonic measure of $\Omega $ $Ω$ , from which a new surface-area-type measure of $\Omega $ $Ω$ is derived. This further motivates us to study the Minkowski problem for this new measure. As a main result, we prove the existence of solutions to the $L_q$ $L_{q}$ Minkowski problem associated to the p-harmonic measure for $0<q<1$ $0 < q < 1$ and $1<p\ne n+1$ $1 < p \neq n + 1$ .

The \(L_q\) Minkowski problem for p-harmonic measure

摘要