A Primer on Semiconvex Functions in General Potential Theories
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Beschreibung
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.
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.
Examines the interplay between nonlinear operator theory and general potential theories Analyzes upper contact jets in the viscosity theory of semiconvex and semicontinuous functions Highlights the monotonicity-duality method for establishing comparison principles
Autor*in
Kevin R. Payne
Themen in »A Primer on Semiconvex Functions in General Potential Theories«
Semiconvexity Upper contact jets Subequation constraint sets Generalized subharmonics Viscosity solutions Comparison principles Fully nonlinear degenerate elliptic operators Partial Differential Equations
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“This book is suitable for a broad audience, including professional mathematicians working in fully nonlinear PDEs, as well as master's and doctoral students with an interest in mathematical analysis. It collects a wealth of results, both classical and new, on pointwise second-order differentiability for functions that are merely semicontinuous, and on viscosity subharmonic functions in nonlinear potential theory. The presentation is fully self-contained and requires only familiarity with basic mathematical analysis, such as multivariable differential calculus and Lebesgue integration theory.” (Juha K. Kinnunen, Mathematical Reviews, May, 2026)
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