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A primer on semiconvex functions in general potential theories

Record Type:	Language materials, printed : Monograph/item
Title/Author:	A primer on semiconvex functions in general potential theories/ by Kevin R. Payne, Davide Francesco Redaelli.
Author:	Payne, Kevin R.
other author:	Redaelli, Davide Francesco.
Published:	Cham :Springer Nature Switzerland : : 2025.,
Description:	xx, 141 p. :ill., digital ; : 24 cm.;
Contained By:	Springer Nature eBook
Subject:	Convex functions. -
Online resource:	https://doi.org/10.1007/978-3-031-94340-9
ISBN:	9783031943409

A primer on semiconvex functions in general potential theories
Payne, Kevin R.

A primer on semiconvex functions in general potential theories[electronic resource] /by Kevin R. Payne, Davide Francesco Redaelli. - Cham :Springer Nature Switzerland :2025. - xx, 141 p. :ill., digital ;24 cm. - Lecture notes in mathematics,v. 23711617-9692 ;. - Lecture notes in mathematics ;1943..

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.

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.

ISBN: 9783031943409

Standard No.: 10.1007/978-3-031-94340-9doiSubjects--Topical Terms:

527742
Convex functions.

LC Class. No.: QA331.5

Dewey Class. No.: 515.8

A primer on semiconvex functions in general potential theories
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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.
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