This book is intended for students who wish to get an introduction to the theory of partial differential equations. The author focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. These are maximum principle methods (particularly important for numerical analysis schemes), parabolic equations, variational methods, and continuity methods. This book also develops the main methods for obtaining estimates for solutions of elliptic equations: Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. Connections between elliptic, parabolic, and hyperbolic equations are explored, as well as the connection with Brownian motion and semigroups. This book can be utilized for a one-year course on partial differential equations. Jürgen Jost is Director of the Max Planck Institute for Mathematics in the Sciences and Professor of Mathematics at the University of Leipzig. He is the author of a number of Springer books, including Postmodern Analysis (1998), Compact Riemann Surfaces (1997) and Riemannian Geometry and Geometric Analysis (1995). The present book is an expanded translation of the original German version, Partielle Differentialgleichungen (1998).

Partial Differential Equations (Graduate Texts in Mathematics)

Science & Math

Jost, Jürgen

Mathematics

Springer

This monograph treats variational problems for mappings from a surface equipped with a conformal structure into Euclidean space or a Riemannian manifold. Presents a general theory of such variational problems, proving existence and regularity theorems with particular conceptual emphasis on the geometric aspects of the theory and thorough investigation of the connections with complex analysis. Among the topics covered are: Plateau's problem, the regularity theory of solutions, a variational approach for obtaining various conformal representation theorems, a general existence theorem for harmonic mappings, and a new approach to Teichmuller theory via harmonic maps.

Two-Dimensional Geometric Variational Problems (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

wiley

Calculus of Variations (Cambridge Studies in Advanced Mathematics, Series Number 64)

This textbook on the calculus of variations leads the reader from the basics to modern aspects of the theory. One-dimensional problems and the classical issues such as Euler-Lagrange equations are treated, as are Noether's theorem, Hamilton-Jacobi theory, and in particular geodesic lines, thereby developing some important geometric and topological aspects. The basic ideas of optimal control theory are also given. The second part of the book deals with multiple integrals. After a review of Lebesgue integration, Banach and Hilbert space theory and Sobolev spaces (with complete and detailed proofs), there is a treatment of the direct methods and the fundamental lower semicontinuity theorems. Subsequent chapters introduce the basic concepts of the modern calculus of variations, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais-Smale condition. The prerequisites are knowledge of the basic results from calculus of one and several variables. After having studied this book, the reader will be well equipped to read research papers in the calculus of variations.

Cambridge University Press

Classical string theory is concerned with the propagation of classical one-dimensional curves, i.e. "strings", and has connections to the calculus of variations, minimal surfaces and harmonic maps. The quantization of string theory gives rise to problems in different areas, according to the method used. The representation theory of Lie, Kac-Moody and Virasoro algebras has been used for such quantization. In this book, the authors give an introduction to global analytic and probabilistic aspects of string theory, bringing together and making explicit the necessary mathematical tools. Researchers with an interest in string theory, in either mathematics or theoretical physics, will find this a stimulating volume.

A Mathematical Introduction to String Theory: Variational Problems, Geometric and Probabilistic Methods (London Mathematical Society Lecture Note Series 225)

Albeverio, Sergio

Paycha, Sylvie

Scarlatti, Sergio

Calculus of Variations

Li-Jost, Xianqing

Presented in this book is a mathematical treatment of Bosonic string theory from the point of view of global geometry. As motivation, the author presents the theory of point particles and Feynman path integrals. He considers the theory of strings as a quantization of the classical Plateau problem for minimal surfaces. The conformal variance of the relevant functional, the Polyakov action or (in mathematical terminology) the Dirichlet integral, leads to an anomaly in the process of quantization. The mathematical concepts needed to resolve this anomaly via the Faddeev-Popov method are introduced, specifically the geometry of the Teichmuüller and moduli spaces of Riemann surfaces and the corresponding function spaces, i.e., Hilbert spaces of Sobolev type and diffeomorphism groups. Other useful tools are the algebraic geometry of Riemann surfaces and infinite-dimensional determinants. Also discussed are the boundary regularity questions. The main result is a presentation of the string partition function as an integral over a moduli space of Riemann surfaces. Some new physical concepts, such as D-branes, are also discussed. This volume offers a mathematically rigorous treatment of some aspects of string theory, employs a global geometry approach, systematically treats strings with boundary, and carefully explains all mathematical concepts and tools.

Bosonic Strings: A Mathematical Treatment (AMS/IP STUDIES IN ADVANCED MATHEMATICS)

Physics

Amer Mathematical Society

This book develops the important approaches to the existence and regularity of partial differential equations, in particular of elliptic (and parabolic) type, particularly those aspects and methods that are relevant not only for linear, but also for nonlinear equations.

Partial Differential Equations (Graduate Texts in Mathematics, 214)

This book helps readers develop mathematical tools required for modern biology. It offers a survey of mathematics, from stochastic processes to pattern formation, and contains biological examples from the molecular to the evolutionary and ecological levels.

Mathematical Methods in Biology and Neurobiology (Universitext)

Reference

This book offers an ideal graduate-level introduction to the theory of partial differential equations. The first part of the book describes the basic mathematical problems and structures associated with elliptic, parabolic, and hyperbolic partial differential equations, and explores the connections between these fundamental types. Aspects of Brownian motion or pattern formation processes are also presented. The second part focuses on existence schemes and develops estimates for solutions of elliptic equations, such as Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. In particular, the reader will learn the basic techniques underlying current research in elliptic partial differential equations.
This revised and expanded third edition is enhanced with many additional examples that will help motivate the reader. New features include a reorganized and extended chapter on hyperbolic equations, as well as a new chapter on the relations between different types of partial differential equations, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, viscosity solutions for elliptic PDEs, and much more. Also, the new edition contains additional material on systems of elliptic partial differential equations, and it explains in more detail how the Harnack inequality can be used for the regularity of solutions.

See the best price to sell, buy, or rent books by Jost, Jürgen. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by Jost, Jürgen

See the best price to sell, buy, or rent books by Jost, Jürgen. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Partial Differential Equations (Graduate Texts in Mathematics)

Two-Dimensional Geometric Variational Problems (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

Calculus of Variations (Cambridge Studies in Advanced Mathematics, Series Number 64)

A Mathematical Introduction to String Theory: Variational Problems, Geometric and Probabilistic Methods (London Mathematical Society Lecture Note Series 225)

Calculus of Variations

Calculus of Variations (Cambridge Studies in Advanced Mathematics, Series Number 64)

Bosonic Strings: A Mathematical Treatment (AMS/IP STUDIES IN ADVANCED MATHEMATICS)

Partial Differential Equations (Graduate Texts in Mathematics, 214)

Mathematical Methods in Biology and Neurobiology (Universitext)

Partial Differential Equations (Graduate Texts in Mathematics, 214)