A primer on quasi-convex functions in nonlinear potential theories

Kevin R. Payne, Davide F. Redaelli

Published 2023-03-25Version 1

We present a self-contained introduction to the fundamental role that quasi-convex functions play in general (nonlinear second order) potential theories, which concerns the study of generalized subharmonics associated to a suitable closed subset of the space of 2-jets. Quasi-convex functions build a bridge between classical and viscosity notions of solutions of the natural Dirichlet problem in any potential theory. Moreover, following a program initiated by Harvey and Lawson in 2009, a potential theoretic-approach is widely being applied for treating nonlinear partial differential equations (PDEs). This viewpoint revisits the conventional viscosity approach to nonlinear PDEs under a more geometric prospective inspired by Krylov and takes much insight from classical pluripotential theory. The possibility of a symbiotic and productive relationship between general potential theories and nonlinear PDEs relies heavily on the class of quasi-convex functions, which are themselves the protagonists of a particular (and important) pure second order potential theory.