Geometric Analysis of Quasilinear Inequalities on Complete Manifolds

Geometric Analysis of Quasilinear Inequalities on Complete Manifolds
Title Geometric Analysis of Quasilinear Inequalities on Complete Manifolds PDF eBook
Author Bruno Bianchini
Publisher Springer Nature
Pages 291
Release 2021-01-18
Genre Mathematics
ISBN 3030627047

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This book demonstrates the influence of geometry on the qualitative behaviour of solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among others, from the theory of submanifolds, the authors study classes of coercive elliptic differential inequalities on domains of a manifold M with very general nonlinearities depending on the variable x, on the solution u and on its gradient. The book highlights the mean curvature operator and its variants, and investigates the validity of strong maximum principles, compact support principles and Liouville type theorems. In particular, it identifies sharp thresholds involving curvatures or volume growth of geodesic balls in M to guarantee the above properties under appropriate Keller-Osserman type conditions, which are investigated in detail throughout the book, and discusses the geometric reasons behind the existence of such thresholds. Further, the book also provides a unified review of recent results in the literature, and creates a bridge with geometry by studying the validity of weak and strong maximum principles at infinity, in the spirit of Omori-Yau’s Hessian and Laplacian principles and subsequent improvements.

Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs

Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs
Title Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs PDF eBook
Author Emanuel Indrei
Publisher American Mathematical Society
Pages 148
Release 2023-01-09
Genre Mathematics
ISBN 147046652X

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This volume contains the proceedings of the virtual conference on Geometric and Functional Inequalities and Recent Topics in Nonlinear PDEs, held from February 28–March 1, 2021, and hosted by Purdue University, West Lafayette, IN. The mathematical content of this volume is at the intersection of viscosity theory, Fourier analysis, mass transport theory, fractional elliptic theory, and geometric analysis. The reader will encounter, among others, the following topics: the principal-agent problem; Maxwell's equations; Liouville-type theorems for fully nonlinear elliptic equations; a doubly monotone flow for constant width bodies; and the edge dislocations problem for crystals that describes the equilibrium configurations by a nonlocal fractional Laplacian equation.

The $AB$ Program in Geometric Analysis: Sharp Sobolev Inequalities and Related Problems

The $AB$ Program in Geometric Analysis: Sharp Sobolev Inequalities and Related Problems
Title The $AB$ Program in Geometric Analysis: Sharp Sobolev Inequalities and Related Problems PDF eBook
Author Olivier Druet
Publisher American Mathematical Soc.
Pages 113
Release 2002
Genre Mathematics
ISBN 0821829890

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Function theory and Sobolev inequalities have been the target of investigation for many years. Sharp constants in these inequalities constitute a critical tool in geometric analysis. The $AB$ programme is concerned with sharp Sobolev inequalities on compact Riemannian manifolds. This text summarizes the results of contemporary research and gives an up-to-date report on the field.

Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs

Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs
Title Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs PDF eBook
Author Alexander Grigor'yan
Publisher Walter de Gruyter GmbH & Co KG
Pages 337
Release 2021-01-18
Genre Mathematics
ISBN 3110700859

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The book covers the latest research in the areas of mathematics that deal the properties of partial differential equations and stochastic processes on spaces in connection with the geometry of the underlying space. Written by experts in the field, this book is a valuable tool for the advanced mathematician.

Geometric Inequalities

Geometric Inequalities
Title Geometric Inequalities PDF eBook
Author Yurii D. Burago
Publisher Springer Science & Business Media
Pages 346
Release 2013-03-14
Genre Mathematics
ISBN 3662074419

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A 1988 classic, covering Two-dimensional Surfaces; Domains on the Plane and on Surfaces; Brunn-Minkowski Inequality and Classical Isoperimetric Inequality; Isoperimetric Inequalities for Various Definitions of Area; and Inequalities Involving Mean Curvature.

Maximum Principles and Geometric Applications

Maximum Principles and Geometric Applications
Title Maximum Principles and Geometric Applications PDF eBook
Author Luis J. Alías
Publisher Springer
Pages 594
Release 2016-02-13
Genre Mathematics
ISBN 3319243373

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This monograph presents an introduction to some geometric and analytic aspects of the maximum principle. In doing so, it analyses with great detail the mathematical tools and geometric foundations needed to develop the various new forms that are presented in the first chapters of the book. In particular, a generalization of the Omori-Yau maximum principle to a wide class of differential operators is given, as well as a corresponding weak maximum principle and its equivalent open form and parabolicity as a special stronger formulation of the latter. In the second part, the attention focuses on a wide range of applications, mainly to geometric problems, but also on some analytic (especially PDEs) questions including: the geometry of submanifolds, hypersurfaces in Riemannian and Lorentzian targets, Ricci solitons, Liouville theorems, uniqueness of solutions of Lichnerowicz-type PDEs and so on. Maximum Principles and Geometric Applications is written in an easy style making it accessible to beginners. The reader is guided with a detailed presentation of some topics of Riemannian geometry that are usually not covered in textbooks. Furthermore, many of the results and even proofs of known results are new and lead to the frontiers of a contemporary and active field of research.

Some Nonlinear Problems in Riemannian Geometry

Some Nonlinear Problems in Riemannian Geometry
Title Some Nonlinear Problems in Riemannian Geometry PDF eBook
Author Thierry Aubin
Publisher Springer Science & Business Media
Pages 414
Release 2013-03-09
Genre Mathematics
ISBN 3662130068

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This book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, geometric and topological methods. It explores important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved.