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The monograph is devoted to the use of the moduli method in mapping theory, in particular, the meaning of direct and inverse modulus inequalities and their possible applications. The main goal is the development of a modulus technique in the Euclidean space and some metric spaces (manifolds, surfaces, quotient spaces, etc.). Particular attention is paid to the local and boundary behavior of mappings, as well as to obtaining modulus inequalities for some classes. The reader is invited to familiarize himself with all the main achievements of the author, synthesized in this book. The results presented here are of a high scientific level, are new and have no analogues in the world with such a degree of generality.

Functions of complex variables. --- Potential theory (Mathematics). --- Functions of a Complex Variable. --- Potential Theory. --- Funcions de variables complexes --- Teoria del potencial (Matemàtica)

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"In this monograph, Cirant et al. prove comparison principles for nonlinear potential theories in Euclidian spaces in a straightforward manner from duality and monotonicity. They also show how to deduce comparison principles for nonlinear differential operators--a program seemingly different from the first. However, this monograph marries these two points of view, for a wide variety of equations, under something called the correspondence principle. Making this connection between potential theory and operator theory enables simplifications on the operator side and provides enrichment on the potential side. Harvey and Lawson have worked for 15 years to articulate a geometric approach to viscosity solutions for an important class of differential equations. Their approach is broader and more flexible than existing alternatives. With the collaboration of Cirant and Payne, this concise book establishes the keystone of the theory: the existence of comparison principles"-- "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"--

Potential theory (Mathematics) --- Differential equations, Partial. --- Nonlinear operators. --- MATHEMATICS / Differential Equations / General --- Annals of Mathematics Studies. --- Comparison Principles for General Potential Theories and PDEs: (AMS-218). --- H. Blaine Lawson, Jr. --- Kevin R. Payne. --- Marco Cirant, F. Reese Harvey. --- Princeton University Press: math. --- Princeton. --- Subequation constraints. --- degenerate elliptic. --- fully nonlinear. --- mathematical theories. --- viscosity solutions.