Interplay between nonlinear potential theory and fully nonlinear elliptic PDEs

TL;DR

This paper explores the deep connection between nonlinear potential theory and fully nonlinear elliptic PDEs, establishing fundamental principles like comparison, existence, and uniqueness of solutions in geometric contexts.

Contribution

It demonstrates how potential theoretic methods can be applied to analyze fully nonlinear elliptic PDEs, especially in geometric settings lacking natural operators.

Findings

Validity of the comparison principle for nonlinear elliptic PDEs

Existence and uniqueness of solutions on pseudoconvex domains

Potential theory provides structural insights into PDE behavior

Abstract

We discuss one of the many topics that illustrate the interaction of Blaine Lawson's deep geometric and analytic insights. The first author is extremely grateful to have had the pleasure of collaborating with Blaine over many enjoyable years. The topic to be discussed concerns the fruitful interplay between nonlinear potential theory; that is, the study of subharmonics with respect to a general constraint set in the 2-jet bundle and the study of subsolutions and supersolutions of a nonlinear (degenerate) elliptic PDE. The main results include (but are not limited to) the validity of the comparison principle and the existence and uniqueness to solutions to the relevant Dirichlet problems on domains which are suitably "pseudoconvex". The methods employed are geometric and flexible as well as being very general on the potential theory side, which is interesting in its own right. Moreover, in many important geometric contexts no natutral operator may be present. On the other hand, the potential theoretic approach can yield results on the PDE side in terms of non standard structual conditions on a given differential operator.