A Primer on Semiconvex Functions in General Potential Theories
Series: Lecture Notes in Mathematics; 2371;
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Product details:
- Publisher Springer Nature Switzerland
- Date of Publication 7 August 2025
- Number of Volumes 1 pieces, Book
- ISBN 9783031943393
- Binding Paperback
- No. of pages141 pages
- Size 235x155 mm
- Language English
- Illustrations XX, 141 p. 3 illus. Illustrations, black & white 685
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Long description:
"
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.
" MoreTable of Contents:
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.
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