Partial Differential Equations

ISBN-13: 9783031466175 / Angielski / Twarda / 2023

Emmanuele DiBenedetto; Ugo Gianazza

Opis

This is a revised and extended version of my 1995 elementary introduction to partial di?erential equations. The material is essentially the same except for three new chapters. The ?rst (Chapter 8) is about non-linear equations of ?rst order and in particular Hamilton–Jacobi equations. It builds on the continuing idea that PDEs, although a branch of mathematical analysis, are closely related to models of physical phenomena. Such underlying physics in turn provides ideas of solvability. The Hopf variational approach to the Cauchy problem for Hamilton–Jacobi equations is one of the clearest and most incisive examples of such an interplay. The method is a perfect blend of classical mechanics, through the role and properties of the Lagrangian and Hamiltonian, and calculus of variations. A delicate issue is that of identifying “uniqueness classes. ” An e?ort has been made to extract the geometrical conditions on the graph of solutions, such as quasi-concavity, for uniqueness to hold. Chapter 9 is an introduction to weak formulations, Sobolev spaces, and direct variationalmethods for linear and quasi-linearelliptic equations. While terse, the material on Sobolev spaces is reasonably complete, at least for a PDEuser. Itincludesallthebasicembeddingtheorems,includingtheirproofs, and the theory of traces. Weak formulations of the Dirichlet and Neumann problems build on this material. Related variational and Galerkin methods, as well as eigenvalue problems, are presented within their weak framework.

Spis treści

Preliminaries.- Quasi-Linear Equations and the Cauchy-Kowalewski Theorem.- The Laplace Equation.- Boundary Value Problems by Double-Layer Potentials.- Integral Equations and Eigenvalue Problems.- The Heat Equation.- The Wave Equation.- Quasi-Linear Equations of First Order.- Linear Elliptic Equations with Measurable Coefficients.- Elliptic De Giorgi Classes.- Navier-Stokes Equations.- Quasi-Linear Hyperbolic First Order Systems.- Non-Linear Equations of the First Order.

O autorze

Emanuele DiBenedetto (1947-2021) was Emeritus Centennial Professor of Mathematics at Vanderbilt University. His teaching and research at Vanderbilt focused on the study of partial differential equations, particularly the elliptic and parabolic ones. He was a prolific writer, authoring more than 120 papers and six books. He served as editor-in-chief of the Journal on Mathematical Analysis of the Society for Industrial and Applied Mathematics. In addition, he was on more than twenty editorial boards and gave numerous invited talks.

Ugo Gianazza is Professor of Mathematical Analysis at the Department of Mathematics F. Casorati of the University of Pavia. His research interests include Regularity for Elliptic and Parabolic Equations. In addition to editing numerous volumes, he also co-authored the 2012 monograph Harnack's Inequality for Degenerate and Singular Parabolic Equation.

Blurb

This graduate textbook provides a self-contained introduction to the classical theory of partial differential equations (PDEs). Through its careful selection of topics and engaging tone, readers will also learn how PDEs connect to cutting-edge research and the modeling of physical phenomena. The scope of the Third Edition greatly expands on that of the previous editions by including five new chapters covering additional PDE topics relevant for current areas of active research. This includes coverage of linear parabolic equations with measurable coefficients, parabolic DeGiorgi classes, Navier-Stokes equations, and more. The “Problems and Complements” sections have also been updated to feature new exercises, examples, and hints toward solutions, making this a timely resource for students.

Partial Differential Equations: Third Edition is ideal for graduate students interested in exploring the theory of PDEs and how they connect to contemporary research. It can also serve as a useful tool for more experienced readers who are looking for a comprehensive reference.