A Minkowski-type problem for measure associated to A-harmonic PDEs

Abstract: 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 associated with a compact convex set E with nonempty interior and its A-harmonic capacitary function in the complement of E. Here A-harmonic PDE is a non-linear elliptic PDE whose structure is modelled on the p-Laplace equation. If \mu_E denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel measure \mu on S^(n-1), find necessary and sufficient conditions for which there exists E as above with \mu_E =\mu. We will discuss the existence, uniqueness, and regularity of this problem in this setting.

analysis of PDEsclassical analysis and ODEsfunctional analysismetric geometry

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.