
Comparison Principles for General Potential Theories and PDEs
Princeton University Press
Published on 3. October 2023
Book
Paperback/Softback
224 pages
978-0-691-24362-7 (ISBN)
Description
An examination of the symbiotic and productive relationship between fully nonlinear partial differential equations and generalized potential theories
In recent years, there has evolved a symbiotic and productive relationship between fully nonlinear partial differential equations and generalized potential theories. This book examines important aspects of this story. One main purpose is to prove comparison principles for nonlinear potential theories in Euclidian spaces straightforwardly from duality and monotonicity under the weakest possible notion of ellipticity. The book also shows how to deduce comparison principles for nonlinear differential operators, by marrying these two points of view, under the correspondence principle.
The authors explain that comparison principles are fundamental in both contexts, since they imply uniqueness for the Dirichlet problem. When combined with appropriate boundary geometries, yielding suitable barrier functions, they also give existence by Perron's method. There are many opportunities for cross-fertilization and synergy. In potential theory, one is given a constraint set of 2-jets that determines its subharmonic functions. The constraint set also determines a family of compatible differential operators. Because there are many such operators, potential theory strengthens and simplifies the operator theory. Conversely, the set of operators associated with the constraint can influence the potential theory.
In recent years, there has evolved a symbiotic and productive relationship between fully nonlinear partial differential equations and generalized potential theories. This book examines important aspects of this story. One main purpose is to prove comparison principles for nonlinear potential theories in Euclidian spaces straightforwardly from duality and monotonicity under the weakest possible notion of ellipticity. The book also shows how to deduce comparison principles for nonlinear differential operators, by marrying these two points of view, under the correspondence principle.
The authors explain that comparison principles are fundamental in both contexts, since they imply uniqueness for the Dirichlet problem. When combined with appropriate boundary geometries, yielding suitable barrier functions, they also give existence by Perron's method. There are many opportunities for cross-fertilization and synergy. In potential theory, one is given a constraint set of 2-jets that determines its subharmonic functions. The constraint set also determines a family of compatible differential operators. Because there are many such operators, potential theory strengthens and simplifies the operator theory. Conversely, the set of operators associated with the constraint can influence the potential theory.
More details
Series
Language
English
Place of publication
New Jersey
United States
Target group
College/higher education
Professional and scholarly
Product notice
Paperback (trade)
Dimensions
Height: 241 mm
Width: 165 mm
Thickness: 36 mm
Weight
363 gr
ISBN-13
978-0-691-24362-7 (9780691243627)
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Schweitzer Classification
Other editions
Additional editions

Marco Cirant | F. Reese Harvey | H. Blaine Lawson
Comparison Principles for General Potential Theories and PDEs
(AMS-218)
E-Book
10/2023
1st Edition
Princeton University Press
€73.49
Available for download
Persons
Marco Cirant is associate professor of mathematical analysis at the Universita di Padova. F. Reese Harvey is professor emeritus of mathematics at Rice University and the author of Spinors and Calibrations. H. Blaine Lawson is distinguished professor of mathematics at Stony Brook University, a member of the National Academy of Sciences, and the author of six books, including Spin Geometry, with Marie-Louise Michelsohn (Princeton). Kevin R. Payne is associate professor of mathematical analysis at the Universita di Milano and a Fellow of the American Mathematical Society.