A primer on semiconvex functions in general potential theories

Abstract

This work is dedicated to foundational aspects of general (nonlinear second order) potential theories and fully nonlinear elliptic PDEs. In particular, we systematically develop the fundamental role played by semiconvex functions as a bridge between the classical and viscosity theory of generalized subharmonics determined by a given subequation constraint set in the bundle of $2$-jets over open subsets of Euclidian spaces. The first part is dedicated to four fundamental and complementary aspects of semiconvex analysis for the viscosity theory of subharmonic functions in general potential theories: differentiability properties of first and second order for locally semiconvex functions; a detailed analysis of their upper contact jets and upper contact points; the deep analytical results which ensure the existence of sets of contact points with positive measure; the well-known device of semiconvex approximation of upper semicontinuous functions. The second part is dedicated to general potential theories. The fundamental roles of duality and monotonicity are discussed, along with the main tools in the viscosity theory of subharmonics and the essential role that semiconvex functions play in that theory. The validity of the comparison principle is examined in two distinct regimes of sufficient monotonicity and minimal monotonicity.