28 feb 2025 -- 12:00   [open in google calendar]

Centro de Giorgi, Sala Conferenze

Abstract.

The classical Minkowski problem is a fundamental inverse problem in convex geometry concerning the prescription of the surface area measure of a convex body, which was asked by Hermann Minkowski in 1897. This problem has been a key inspiration in the study of the fully nonlinear PDEs, as in the smooth setting, it reduces to a Monge-Ampère equation on the unit sphere. Through the seminal works of Nirenberg, Pogorelov, and Cheng-Yau, among many others, this problem has been fully resolved and was a milestone in global geometry. In this talk, we will discuss a boundary value problem of the classical Minkowski problem, which concerns the existence of a convex hypersurface with prescribed Gauss-Kronecker curvature and a capillary boundary supported on an obstacle. By formulating this as a Monge-Ampère equation with a Robin (or Neumann) boundary condition on a spherical cap, we establish the existence of smooth solutions to this problem.