In this article we study two classical problems in convex geometry associated to A-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modeled on the p-Laplace equation. Let p be fixed with 2 ≤ n ≤ p < ∞. For a convex compact set E in R, 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 BrunnMinkowski type inequality [CA(λE1 + (1− λ)E2)] 1 p−n ≥ λ [CA(E1)] 1 p−n + (1− λ) [CA(E2)] 1 p−n when n < p <∞, 0 ≤ λ ≤ 1, and E1, E2 are nonempty convex compact sets in R. While 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 R 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 Sn−1 to be the surface area measure of a convex compact set in R 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. 2010 Mathematics Subject Classification. 35J60,31B15,39B62,52A40,35J20,52A20,35J92.

中文翻译：

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Brunn--Minkowski 不等式和 𝒜-调和格林函数的 Minkowski 问题

在本文中，我们研究了与 A 谐波偏微分方程相关的凸几何中的两个经典问题，即准线性椭圆偏微分方程，其结构以 p-拉普拉斯方程为模型。让 p 固定为 2 ≤ n ≤ p < ∞。对于 R 中的凸紧集 E，我们定义并证明所谓的 A-调和格林函数的存在性和唯一性，即 E 的补集在无穷远处。然后我们定义一个量 CA(E)，它可以被看作是这个函数接近无穷大的行为。在本文的第一部分，我们证明 CA(·) 满足以下 BrunnMinkowski 型不等式 [CA(λE1 + (1− λ)E2)] 1 p−n ≥ λ [CA(E1)] 1 p−n + (1− λ) [CA(E2)] 1 p−n 当 n < p <∞, 0 ≤ λ ≤ 1，并且 E1, E2 是 R 中的非空凸紧集。当 p = n 时，CA(λE1 + (1− λ)E2) ≥ λCA(E1) + (1− λ)CA(E2) 其中 0 ≤ λ ≤ 1 和 E1，E2 是 R 中至少包含两个点的凸紧致集。此外，如果对于某些 E1 和 E2 在上述不等式中的任何一个中相等，那么在 A 的某些规则性和结构假设下，我们证明这两个集合是同位的。经典的 Minkowski 问题要求在单位球面 Sn-1 上的非负 Borel 测度上的充分必要条件是在该凸集边界的高斯映射下 R 中凸紧凑集的表面积测度。在本文的第二部分中，我们研究了一个 Minkowski 类型问题，该问题与当 n ≤ p < ∞ 时凸紧集 E 的补集的 A 谐波格林函数相关联。如果 μE 表示这个度量，然后我们证明在这种设置下存在的充分必要条件与经典闵可夫斯基问题中的条件完全相同。使用第一部分的 Brunn-Minkowski 不等式结果，我们还表明这个问题在翻译之前有一个独特的解决方案。2010年数学学科分类。35J60、31B15、39B62、52A40、35J20、52A20、35J92。