A Primer on Semiconvex Functions in General Potential Theories
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, i...
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| Hoofdauteurs: | , |
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| Coauteur: | |
| Formaat: | Elektronisch E-boek |
| Taal: | Engels |
| Gepubliceerd in: |
Cham :
Springer Nature Switzerland : Imprint: Springer,
2025.
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| Editie: | 1st ed. 2025. |
| Reeks: | Lecture Notes in Mathematics,
2371 |
| Onderwerpen: | |
| Online toegang: | https://doi.org/10.1007/978-3-031-94340-9 |
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| Samenvatting: | 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. |
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| Fysieke beschrijving: | 1 online resource (XX, 141 p. 3 illus.). |
| ISBN: | 9783031943409 |
| ISSN: | 1617-9692 ; |
| DOI: | 10.1007/978-3-031-94340-9 |