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 $\cF$ 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 $F$ acting on the space of 2-jets. We will discuss a powerful tool, the correspondence principle, which establishes the equivalence between $\cF$–subharmonics/superharmonics $u$ and admissible subsolutions/supersolutions $u$ (in the viscosity sense) of the PDE determined by every operator $F$ which is compatible with $\cF$. The crucial degenerate ellipticity often requires the operator to be restricted to a suitable constraint set $\cG$, which determines the admissibility. Applications to comparison principles by way of the duality-monotonicity-fiberegularity method will also be discussed.