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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathcal{A}} -harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p -Laplace equation. Let p be fixed with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>≤</m:mo> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> {2\leq n\leq p&lt;\infty} . For a convex compact set E in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {\mathbb{R}^{n}} , we define and then prove the existence and uniqueness of the so-called <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒜</m:mi> </m:math> {\mathcal{A}} -harmonic Green’s function for the complement of E with pole at infinity. We then define a quantity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">C</m:mi> <m:mi mathvariant="script">𝒜</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>E</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="normal">C</m:mi> <m:mi mathvariant="script">𝒜</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">(</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{C}_{\mathcal{A}}(\,\cdot\,)} satisfies the following Brunn–Minkowski-type inequality: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">C</m:mi> <m:mi mathvariant="script">𝒜</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>E</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mi>λ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msub> <m:mi>E</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:msup> <m:mo>≥</m:mo> <m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="normal">C</m:mi> <m:mi mathvariant="script">𝒜</m:mi> </m:msub> <m:mo>⁢</m:mo>