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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| Natura: | Elettronico eBook |
| Lingua: | inglese |
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Cham :
Springer Nature Switzerland : Imprint: Springer,
2025.
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| Edizione: | 1st ed. 2025. |
| Serie: | Lecture Notes in Mathematics,
2371 |
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| Accesso online: | https://doi.org/10.1007/978-3-031-94340-9 |
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| 100 | 1 | |a Payne, Kevin R. |e author. |4 aut |4 http://id.loc.gov/vocabulary/relators/aut | |
| 245 | 1 | 2 | |a A Primer on Semiconvex Functions in General Potential Theories |h [electronic resource] / |c by Kevin R. Payne, Davide Francesco Redaelli. |
| 250 | |a 1st ed. 2025. | ||
| 264 | 1 | |a Cham : |b Springer Nature Switzerland : |b Imprint: Springer, |c 2025. | |
| 300 | |a 1 online resource (XX, 141 p. 3 illus.). | ||
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| 490 | 1 | |a Lecture Notes in Mathematics, |x 1617-9692 ; |v 2371 | |
| 505 | 0 | |a 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. | |
| 520 | |a 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. | ||
| 532 | 8 | |a Accessibility summary: This PDF has been created in accordance with the PDF/UA-1 standard to enhance accessibility. It includes screen reader support, descriptions for Non-text content (such as images and graphs), bookmarks for easy navigation, keyboard-friendly links and forms, and searchable, selectable text. We recognise the importance of accessibility and welcome queries about accessibility for any of our products. If you have a question or an access need, please get in touch with us via https://www.springernature.com/gp/info/accessibility Please Note that a more accessible version of this eBook is available as an EPUB. | |
| 532 | 8 | |a Inaccessible, or known limited accessibility | |
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| 532 | 8 | |a Publisher contact for further accessibility information: accessibilitysupport@springernature.com | |
| 588 | |a Print version record. | ||
| 650 | 0 | |a Functions of real variables. | |
| 650 | 0 | |a Potential theory (Mathematics). | |
| 650 | 0 | |a Differential equations. | |
| 650 | 1 | 4 | |a Real Functions. |
| 650 | 2 | 4 | |a Potential Theory. |
| 650 | 2 | 4 | |a Differential Equations. |
| 700 | 1 | |a Redaelli, Davide Francesco. |e author. |0 (orcid)0009-0004-9125-9022 |1 https://orcid.org/0009-0004-9125-9022 |4 aut |4 http://id.loc.gov/vocabulary/relators/aut | |
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