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This book examines the symbiotic interplay between fully nonlinear elliptic partial differential equations and general potential theories of second order. Starting with a self-contained presentation of the classical theory of first and second order differentiability properties of convex functions, it collects a wealth of results on how to treat second order differentiability in a pointwise manner for merely semicontinuous functions. The exposition features an analysis of upper contact jets for semiconvex functions, a proof of the equivalence of two crucial, independently developed lemmas of Jensen (on the viscosity theory of PDEs) and Slodkowski (on pluripotential theory), and a detailed description of the semiconvex approximation of upper semicontinuous functions.
The foundations of general potential theories are covered, with a review of monotonicity and duality, and the basic tools in the viscosity theory of generalized subharmonics, culminating in an account of the monotonicity-duality method for proving comparison principles. The final section shows that the notion of semiconvexity extends naturally to manifolds. A complete treatment of important background results, such as Alexandrov s theorem and a Lipschitz version of Sard s lemma, is provided in two appendices.
The book is aimed at a wide audience, including professional mathematicians working in fully nonlinear PDEs, as well as master s and doctoral students with an interest in mathematical analysis.
Table des matières
Part I. Semiconvex apparatus.- Chapter 1. Differentiability of convex functions.- Chapter 2. Semiconvex functions and upper contact jets.- Chapter 3. The lemmas of Jensen and Slodkowski.- Chapter 4. Semiconvex approximation of semicontinuous functions.- Part II. General potential-theoretic analysis.- Chapter 5. General potential theories.- Chapter 6. Duality and monotonicity in general potential theories.- Chapter 7. Basic tools in nonlinear potential theory.- Chapter 8. Semiconvex functions and subharmonics.- Chapter 9. Comparison principles.- Chapter 10. From Euclidean spaces to manifolds: a brief note.
