The Correspondence Principle: A bridge between general potential theories and nonlinear elliptic differential operators

Abstract

General potential theories concern the study of functions which are subharmonic with respect to a suitable constraint set (called a subequation) in the space of 2-jets. While interesting in their own right, general potential theories are being widely used to study fully nonlinear PDEs determined by degenerate elliptic operators acting on the space of 2-jets. We will discuss a powerful tool, the correspondence principle, which establishes the equivalence between subequation subharmonics (superharmonics) and admissible subsolutions (supersolutions) in the viscosity sense of the PDE determined by every operator which is compatible with a given subequation. The crucial degenerate ellipticity often requires the operator to be restricted to a suitable constraint set, which determines the admissibility. Applications to comparison principles by way of the duality monotonicity fiberegularity method will also be discussed.