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Brunn-Minkowsky Inequality for p-Harmonic Measures

Ariel A. Aguas-Barreno, Murat Akman, Shirsho Mukherjee

Abstract

We prove a local Brunn-Minkowski inequality for a functional corresponding to p-harmonic measures for 2 < p < n+1.

Brunn-Minkowsky Inequality for p-Harmonic Measures

Abstract

We prove a local Brunn-Minkowski inequality for a functional corresponding to p-harmonic measures for 2 < p < n+1.
Paper Structure (10 sections, 22 theorems, 180 equations, 1 figure)

This paper contains 10 sections, 22 theorems, 180 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex domains and support functions
  4. The $p$-Laplacian and $p$-harmonic measures
  5. Minkowski problem
  6. The Brunn-Minkowski inequality
  7. Support functions on sub-level sets
  8. Sub-solutions from supremal convolution
  9. Homogeneity and Hadamard's formula
  10. Proof of the Theorem

Key Result

Theorem 1.5

Let $K_0\subset \mathbb{R}^n$ be a compact convex set of non-empty interior and $\omega_{0}$ be any given $p$-harmonic measure on $\partial K_0$ for $2<p<n+1$. There exists a neighborhood $\mathcal{N}$ of convex sets of non-empty interior containing $K_0$ such that for any $K\in \mathcal{N}$, there where $h_K:\mathbb S^{n-1} \to \mathbb{R}$ is the support function of $K$ and $\mathbf{g}_K:\partia

Figures (1)

  • Figure 1:

Theorems & Definitions (41)

  • Theorem 1.5
  • Lemma 2.20
  • Remark 2.21
  • Theorem 2.25
  • Lemma 2.37
  • Theorem 2.39: Existence
  • Remark 2.40
  • Remark 2.41
  • Lemma 3.10
  • proof
  • ...and 31 more