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

In this article we study two classical problems in convex geometry associated to A-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace equation. Let p be fixed with 2≤n≤p<∞. For a convex compact set E in ℝn, we define and then prove the existence and uniqueness of the so-called A-harmonic Green's function for the complement of E with pole at infinity. We then define a quantity CA(E) which can be seen as the behavior of this function near infinity. In the first part of this article, we prove that CA (˙) satisfies the following Brunn-Minkowski-type inequality: [CA(λ E1+(1-λ)E2)]1p-n≥λ[CA(E1)]1p-n+(1-λ)[CA(E2)]1p-n when n<p<∞, 0≤λ≤1, and E1,E2 are nonempty convex compact sets in n. While p=n{p=n} then CA(λE1+(1-λ)E2)≥λCA(E1)+(1-λ)CA(E2), where 0≤λ≤1 and E1,E2 are convex compact sets in ℝn containing at least two points. Moreover, if equality holds in the either of the above inequalities for some E1 and E2, then under certain regularity and structural assumptions on 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 n-1 to be the surface area measure of a convex compact set in ℝn 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-harmonic Green's function for the complement of a convex compact set E when n≤p<∞. If μE 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.