Stability of Brunn-Minkowski inequalities and Minkowski type problems for nonlinear capacity

The origin of potential theory goes back to Newton's work on laws of mechanics in 1687 while studying the properties of forces which follow the law of gravitation. This theory has been widely used during the 17th and 18th centuries by Lagrange, Legendre, Laplace, and Gauss to study problems in the theory of gravitation, electrostatics and magnetism. It was observed that these forces could be modeled using so called harmonic functions which are solutions to a very special linear partial differential equation (PDE) known as Laplace's equation. A measuring notion called capacity appears in Physics and is defined as the ability of a body to hold an electrical charge. Mathematically, it can be calculated in terms of an integral of a certain harmonic function. The capacity has been widely used while studying harmonic functions and this field of Mathematics is called Potential Theory. This theory branched off in many directions including nonlinear potential theory of p-Laplace equation and A-harmonic PDEs. These are second-order elliptic PDEs and can be seen as a nonlinear generalization of Laplace's equation. A-harmonic PDEs have received little attention due to their nonlinearity and recently found applications in rheology, glaciology, radiation of heat, plastic moulding. Nonlinear capacity associated to A-harmonic PDEs naturally appears while studying boundary value problems for A-harmonic PDEs.

A mathematical operation called Minkowski addition of sets appears in convex analysis. It is defined by addition of all possible sums in the sets and it appears in motion planning, 3D solid modeling, aggregation theory, and collision detection. Classical Brunn-Minkowski inequality has been known for more than a century and relates the volumes of subsets of Euclidean space under the Minkowski addition. It has been obtained for various other quantities including capacity obtained by C. Borell. Recently, the PI and his collaborators observed that nonlinear capacity satisfies a Brunn-Minkowski type inequality and it states that a certain power of it is a concave function under the Minkowski addition of any convex compact sets including low-dimensional sets. Inspired by the recent development on stability of the classical Brunn-Minkowski inequality by M. Christ, A. Figalli, and D. Jerison, the first part of this project is devoted to studying the stability of Brunn-Minkowski inequality for nonlinear capacity associated to A-harmonic PDEs for convex compact sets. This is a new and challenging direction of research as this problem has not been addressed even for the Logarithmic or Newtonian capacity associated to Laplacian. The project will also investigate sharpness of these inequalities for non-convex sets.

Once the Brunn-Minkowski inequality has been studied, it is natural to study a related problem which is known as the Minkowski problem. This 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. The PI and his collaborators have studied this problem from the potential theoretic point of view when underlying equations are A-harmonic PDEs and solved the existence and uniqueness in this setting. The second part of the project focuses on regularity of the Minkowski problem for nonlinear capacity associated to A-harmonic PDEs. This requires further work on regularity of solutions to a system of PDEs involving Monge-Ampere equation, a nonlinear second-order PDE of special kind, and A-harmonic PDEs. Building on D. Jerison's work, the project also aims to increase understanding of A-harmonic measures of convex domains associated to A-harmonic PDEs by studying a Minkowski-type problem.