A primer on semiconvex functions in general potential theories / Kevin R. Payne, Davide Francesco Redaelli.

Format:

Book

Author/Creator:

Payne, Kevin R., 1962- author.

Redaelli, Davide Francesco, author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 0075-8434 no. 2371.

Lecture notes in mathematics, 0075-8434 ; 2371

Language:

English

Subjects (All):

Differential equations, Elliptic.

Differential equations, Nonlinear.

Potential theory (Mathematics).

Physical Description:

xx, 139 pages ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer, [2025]

Summary:

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.

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.

Notes:

Includes bibliographical references (pages 129-131) and indexes.

ISBN:

9783031943393

3031943392

OCLC:

1535231736

Publisher Number:

CIPO000284028