### A Minkowski problem for nonlinear capacity

Analysis and Geometry Seminar

25th April 2024, 3:30 pm – 4:30 pm

Fry Building, Room 2.04

The classical Minkowski problem consists in finding a convex polyhedron from data consisting of normals to their faces and their surface areas. In the smooth case, the corresponding problem for convex bodies is to find the convex body given the Gauss curvature of its boundary, as a function of the unit normal. The proof consists of three parts: existence, uniqueness and regularity.

In this talk, we study a Minkowski problem for certain measure, called p-capacitary surface area measure, associated to a compact convex set $E$ with nonempty interior and its $p-$harmonic capacitary function (solution to the p-Laplace equation in the complement of $E$). If $\mu_p$ denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel positive measure $\mu$ on $\mathbb{S}^{n-1}$, find necessary and sufficient conditions for which there exists a convex body $E$ with $\mu_p =\mu$. We will discuss the existence, uniqueness, and regularity of this problem which have deep connections with the Brunn-Minkowski inequality for p-capacity and Monge-Ampère equation.

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