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# Advances in Calculus of Variations

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# The Brunn--Minkowski inequality and a Minkowski problem for 𝒜-harmonic Green's function

Murat Akman
/ John Lewis
/ Olli Saari
/ Andrew Vogel
Published Online: 2019-04-06 | DOI: https://doi.org/10.1515/acv-2018-0064

## Abstract

In this article we study two classical problems in convex geometry associated to $\mathcal{𝒜}$-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace equation. Let p be fixed with $2\le n\le p<\mathrm{\infty }$. For a convex compact set E in ${ℝ}^{n}$, we define and then prove the existence and uniqueness of the so-called $\mathcal{𝒜}$-harmonic Green’s function for the complement of E with pole at infinity. We then define a quantity ${\mathrm{C}}_{\mathcal{𝒜}}\left(E\right)$ which can be seen as the behavior of this function near infinity. In the first part of this article, we prove that ${\mathrm{C}}_{\mathcal{𝒜}}\left(\cdot \right)$ satisfies the following Brunn–Minkowski-type inequality:

${\left[{\mathrm{C}}_{\mathcal{𝒜}}\left(\lambda {E}_{1}+\left(1-\lambda \right){E}_{2}\right)\right]}^{\frac{1}{p-n}}\ge \lambda {\left[{\mathrm{C}}_{\mathcal{𝒜}}\left({E}_{1}\right)\right]}^{\frac{1}{p-n}}+\left(1-\lambda \right){\left[{\mathrm{C}}_{\mathcal{𝒜}}\left({E}_{2}\right)\right]}^{\frac{1}{p-n}}$

when $n, $0\le \lambda \le 1$, and ${E}_{1},{E}_{2}$ are nonempty convex compact sets in ${ℝ}^{n}$. While $p=n$ then

${\mathrm{C}}_{\mathcal{𝒜}}\left(\lambda {E}_{1}+\left(1-\lambda \right){E}_{2}\right)\ge \lambda {\mathrm{C}}_{\mathcal{𝒜}}\left({E}_{1}\right)+\left(1-\lambda \right){\mathrm{C}}_{\mathcal{𝒜}}\left({E}_{2}\right),$

where $0\le \lambda \le 1$ and ${E}_{1},{E}_{2}$ are convex compact sets in ${ℝ}^{n}$ containing at least two points. Moreover, if equality holds in the either of the above inequalities for some ${E}_{1}$ and ${E}_{2}$, then under certain regularity and structural assumptions on $\mathcal{𝒜}$ we show that these two sets are homothetic. The classical Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere ${𝕊}^{n-1}$ to be the surface area measure of a convex compact set in ${ℝ}^{n}$ under the Gauss mapping for the boundary of this convex set. In the second part of this article we study a Minkowski-type problem for a measure associated to the $\mathcal{𝒜}$-harmonic Green’s function for the complement of a convex compact set E when $n\le p<\mathrm{\infty }$. If ${\mu }_{E}$ denotes this measure, then we show that necessary and sufficient conditions for existence under this setting are exactly the same conditions as in the classical Minkowski problem. Using the Brunn–Minkowski inequality result from the first part, we also show that this problem has a unique solution up to translation.

MSC 2010: 35J60; 31B15; 39B62; 52A40; 35J20; 52A20; 35J92

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Revised: 2019-03-03

Accepted: 2019-03-07

Published Online: 2019-04-06

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: DFG-SFB 1060

Funding Source: Division of Mathematical Sciences

Award identifier / Grant number: DMS-1265996

Award identifier / Grant number: DMS-1440140

This material is based upon work supported by National Science Foundation under Grant No. DMS-1440140 while the first and the third authors were in residence at the MSRI in Berkeley, California, during the Spring 2017 semester. The second author was partially supported by NSF DMS-1265996. The third author was partially supported by the Hausdorff Center for Mathematics as well as DFG-SFB 1060.

Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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