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

Abstract

Abstract In this article we study two classical problems in convex geometry associated to 𝒜 {\mathcal{A}} -harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace equation. Let p be fixed with 2 ≤ n ≤ p < ∞ {2\leq n\leq p<\infty} . For a convex compact set E in ℝ n {\mathbb{R}^{n}} , we define and then prove the existence and uniqueness of the so-called 𝒜 {\mathcal{A}} -harmonic Green’s function for the complement of E with pole at infinity. We then define a quantity C 𝒜 ⁢ ( E ) {\mathrm{C}_{\mathcal{A}}(E)} which can be seen as the behavior of this function near infinity. In the first part of this article, we prove that C 𝒜 ⁢ ( ⋅ ) {\mathrm{C}_{\mathcal{A}}(\,\cdot\,)} satisfies the following Brunn–Minkowski-type inequality: [ C 𝒜 ⁢ ( λ ⁢ E 1 + ( 1 - λ ) ⁢ E 2 ) ] 1 p - n ≥ λ ⁢ [ C 𝒜 ⁢ ( E 1 ) ] 1 p - n + ( 1 - λ ) ⁢ [ C 𝒜 ⁢ ( E 2 ) ] 1 p - n [\mathrm{C}_{\mathcal{A}}(\lambda E_{1}+(1-\lambda)E_{2})]^{\frac{1}{p-n}}\geq% \lambda[\mathrm{C}_{\mathcal{A}}(E_{1})]^{\frac{1}{p-n}}+(1-\lambda)[\mathrm{C% }_{\mathcal{A}}(E_{2})]^{\frac{1}{p-n}} when n < p < ∞ {n<p<\infty} , 0 ≤ λ ≤ 1 {0\leq\lambda\leq 1} , and E 1 , E 2 {E_{1},E_{2}} are nonempty convex compact sets in ℝ n {\mathbb{R}^{n}} . While p = n {p=n} then C 𝒜 ⁢ ( λ ⁢ E 1 + ( 1 - λ ) ⁢ E 2 ) ≥ λ ⁢ C 𝒜 ⁢ ( E 1 ) + ( 1 - λ ) ⁢ C 𝒜 ⁢ ( E 2 ) , \mathrm{C}_{\mathcal{A}}(\lambda E_{1}+(1-\lambda)E_{2})\geq\lambda\mathrm{C}_% {\mathcal{A}}(E_{1})+(1-\lambda)\mathrm{C}_{\mathcal{A}}(E_{2}), where 0 ≤ λ ≤ 1 {0\leq\lambda\leq 1} and E 1 , E 2 {E_{1},E_{2}} are convex compact sets in ℝ n {\mathbb{R}^{n}} containing at least two points. Moreover, if equality holds in the either of the above inequalities for some E 1 {E_{1}} and E 2 {E_{2}} , then under certain regularity and structural assumptions on 𝒜 {\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 {\mathbb{S}^{n-1}} to be the surface area measure of a convex compact set in ℝ n {\mathbb{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 𝒜 {\mathcal{A}} -harmonic Green’s function for the complement of a convex compact set E when n ≤ p < ∞ {n\leq p<\infty} . If μ E {\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.