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arXiv · 2412.07561

The $L_q$ Minkowski problem for $\mathbf{p}$-harmonic measure

Hai Li·Longyu Wu·Baocheng Zhu
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Abstract

In this paper, we consider an extremal problem associated with the solution to a boundary value problem. Our main focus is on establishing a variational formula for a functional related to the $\mathbf{p}$-harmonic measure, from which a new measure 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 $L_q$ Minkowski problem associated with the $\mathbf{p}$-harmonic measure for $0<q<1$ and $1<\mathbf{p}\ne n+1$.

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Hai Li, Longyu Wu, Baocheng Zhu. 2024-12-10. The $L_q$ Minkowski problem for $\mathbf{p}$-harmonic measure. https://arxiv.org/abs/2412.07561

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