Boundary regularity of Riesz potential, smooth solution to the chord log-Minkowski problem

Abstract

The Riesz potential and its potential theory are closely related to the regularity of solutions to partial differential equations. In this paper, we investigate a class of Minkowski type problems that are closely associated with convex geometry and geometric probability. This investigation leads to a novel result regarding the boundary regularity of the Riesz potential, which is essential for developing the regularity theory of a new nonlocal {M}onge-{A}mp\`ere equation. As an initial application, the existence of a smooth solution to the chord log-Minkowski problem shall be obtained from the perspective of a nonlocal elliptical partial differential equation and a nonlocal Gauss curvature flow equation.