Comparison Principles for General Potential Theories and PDEs: (AMS-218)

Format: Paperback / softback
Length: 224 pages
Publication date: 03 October 2023
Publisher: Princeton University Press

In recent years, a remarkable symbiotic and fruitful relationship has emerged between fully nonlinear partial differential equations and generalized potential theories. This captivating book delves into the intricate details of this evolving interplay, exploring key aspects of this captivating story. One of the primary objectives of this work is to establish comparison principles for nonlinear potential theories in Euclidian spaces through straightforward derivations from duality and monotonicity, employing the weakest possible notion of ellipticity. Moreover, the book showcases how to deduce comparison principles for nonlinear differential operators by seamlessly merging these two perspectives, leveraging the correspondence principle.

The authors elucidate that comparison principles hold immense significance in both contexts, as they imply uniqueness for the Dirichlet problem. When coupled with appropriate boundary geometries, yielding suitable barrier functions, they also result in the existence of solutions by the Perron method. This interplay presents a multitude of opportunities for cross-fertilization and synergy. In the realm of potential theory, a constraint set of 2-jets is provided, which determines the subharmonic functions of the theory. Furthermore, this constraint set establishes a family of compatible differential operators. Due to the abundance of these operators, potential theory enhances and simplifies the operator theory. Conversely, the set of operators associated with the constraint can influence the potential theory.

This book serves as a valuable resource for researchers and scholars in the fields of mathematics, physics, and engineering, offering a comprehensive exploration of the symbiotic and productive relationship between fully nonlinear partial differential equations and generalized potential theories. By unraveling the complexities and connections between these two areas, the book provides insights into the latest developments and advancements in mathematical research, paving the way for new discoveries and breakthroughs.

Dimension: 235 x 156 (mm)
ISBN-13: 9780691243627