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Lq-Minkowski problem of anisotropic p-torsional rigidity

Chao Li, Bin Chen

TL;DR

This paper addresses the $L_q$-Minkowski problem for anisotropic $p$-torsional rigidity, extending the classical Minkowski framework to the anisotropic $p$-Laplacian setting. It develops a variational approach via the $L_q$-anisotropic $p$-torsional measure $S_{F,p,q}$ and a functional $\Psi_{F,p,q}$ to establish existence results. The authors prove existence for $1<p<\infty$, $1<q\neq \frac{p}{p-1}+n$ under nondegeneracy of the measure, and for $0<q<1$ under discrete-to-general measure approximation, using polyhedral approximations, Blaschke selection, and weak convergence. This work unifies convex geometry and PDE techniques, showing that $S_{F,p,q}(K,\cdot)$ characterizes given measures as torsional-geometry data of convex bodies, with multiple specializations recovering known torsional and Minkowski-type problems. The results broaden the class of Minkowski-type problems available for anisotropic torsional problems and provide a robust variational framework for future extensions.

Abstract

In this paper, the $L_q$-Minkowski problem of anisotropic $p$-torsional rigidity is considered. The existence of the solution of the $L_q$-Minkowski problem of anisotropic $p$-torsional rigidity with $0<q<1$ and $1<q\neq \frac{p}{p-1}+n$ is given.

Lq-Minkowski problem of anisotropic p-torsional rigidity

TL;DR

This paper addresses the -Minkowski problem for anisotropic -torsional rigidity, extending the classical Minkowski framework to the anisotropic -Laplacian setting. It develops a variational approach via the -anisotropic -torsional measure and a functional to establish existence results. The authors prove existence for , under nondegeneracy of the measure, and for under discrete-to-general measure approximation, using polyhedral approximations, Blaschke selection, and weak convergence. This work unifies convex geometry and PDE techniques, showing that characterizes given measures as torsional-geometry data of convex bodies, with multiple specializations recovering known torsional and Minkowski-type problems. The results broaden the class of Minkowski-type problems available for anisotropic torsional problems and provide a robust variational framework for future extensions.

Abstract

In this paper, the -Minkowski problem of anisotropic -torsional rigidity is considered. The existence of the solution of the -Minkowski problem of anisotropic -torsional rigidity with and is given.

Paper Structure

This paper contains 6 sections, 13 theorems, 150 equations.

Table of Contents

  1. Introduction and main results
  2. Notations and Background Materials
  3. Convex Geometry
  4. Anisotropic $p$-torsional rigidity
  5. A variational proof of $L_q$-Minkowski problem of anisotropic $p$-torsional measure
  6. The $L_q$-Minkowski problem of anisotropic $p$-torsional rigidity for general measure with $0<q<1$

Key Result

Theorem 1.1

Let $1<p<\infty$, $1<q\neq \frac{p}{p-1}+n$ and $\mu$ be a nonzero finite Borel measure on $\mathbb{S}^{n-1}$ that is not concentrated in any closed hemisphere, then there exists a convex body $K \in \mathscr{K}_o^n$ such that

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1: Porposition 2.5 of KLL2023
  • Lemma 2.2: Lemma 2.1 of LJQ2022
  • Lemma 2.3: PDG2014
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • ...and 11 more