Author : Stefano Pigola
Publisher : American Mathematical Soc.
Page : 118 pages
File Size : 15,10 MB
Release : 2005
Category : Mathematics
ISBN : 0821836390
Aims to introduce the reader to various forms of the maximum principle, starting from its classical formulation up to generalizations of the Omori-Yau maximum principle at infinity obtained by the authors.
Author : Marco Cirant
Publisher : Princeton University Press
Page : 225 pages
File Size : 30,53 MB
Release : 2023-10-03
Category : Mathematics
ISBN : 0691243646
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.
Author : Bruno Bianchini
Publisher : Springer Nature
Page : 291 pages
File Size : 26,25 MB
Release : 2021-01-18
Category : Mathematics
ISBN : 3030627047
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.
Author : Bruno Bianchini
Publisher : American Mathematical Soc.
Page : 208 pages
File Size : 35,17 MB
Release : 2013-08-23
Category : Mathematics
ISBN : 0821887998
The aim of this paper is to analyze some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. With this motivation the authors prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE's that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep their investigation basically self-contained, the authors also collect some, more or less known, material which often appears in the literature in various forms and for which they give, in some instances, new proofs according to their specific point of view.
Author : Massimo A. Picardello
Publisher : Springer Science & Business Media
Page : 450 pages
File Size : 28,6 MB
Release : 2012-12-05
Category : Mathematics
ISBN : 8847028531
This book illustrates the wide range of research subjects developed by the Italian research group in harmonic analysis, originally started by Alessandro Figà-Talamanca, to whom it is dedicated in the occasion of his retirement. In particular, it outlines some of the impressive ramifications of the mathematical developments that began when Figà-Talamanca brought the study of harmonic analysis to Italy; the research group that he nurtured has now expanded to cover many areas. Therefore the book is addressed not only to experts in harmonic analysis, summability of Fourier series and singular integrals, but also in potential theory, symmetric spaces, analysis and partial differential equations on Riemannian manifolds, analysis on graphs, trees, buildings and discrete groups, Lie groups and Lie algebras, and even in far-reaching applications as for instance cellular automata and signal processing (low-discrepancy sampling, Gaussian noise).
Author : E. Brian Davies
Publisher : Cambridge University Press
Page : 344 pages
File Size : 41,26 MB
Release : 1999-09-30
Category : Mathematics
ISBN : 0521777496
Authoritative lectures from world experts on spectral theory and geometry.
Author : Karsten Grove
Publisher : Cambridge University Press
Page : 280 pages
File Size : 35,62 MB
Release : 1997-05-13
Category : Mathematics
ISBN : 9780521592222
This is an up to date work on a branch of Riemannian geometry called Comparison Geometry.
Author : Emanuel Indrei
Publisher : American Mathematical Society
Page : 148 pages
File Size : 38,98 MB
Release : 2023-01-09
Category : Mathematics
ISBN : 147046652X
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.
Author : Jeff Cheeger
Publisher : Edizioni della Normale
Page : 0 pages
File Size : 29,62 MB
Release : 2001-10-01
Category : Mathematics
ISBN : 9788876423048
These notes are based on the Fermi Lectures delivered at the Scuola Normale Superiore, Pisa, in June 2001. The principal aim of the lectures was to present the structure theory developed by Toby Colding and myself, for metric spaces which are Gromov-Hausdorff limits of sequences of Riemannian manifolds which satisfy a uniform lower bound of Ricci curvature. The emphasis in the lectures was on the “non-collapsing” situation. A particularly interesting case is that in which the manifolds in question are Einstein (or Kähler-Einstein). Thus, the theory provides information on the manner in which Einstein metrics can degenerate.
Author : Steven Rosenberg
Publisher : Cambridge University Press
Page : 190 pages
File Size : 43,15 MB
Release : 1997-01-09
Category : Mathematics
ISBN : 9780521468312
This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.
