偏微分方程的研究生教材中最著名的莫过于Evans的Partial Differential Equations （2nd， GMS19.AMS），内容讲解深入浅出，涉及到PDE的大部分领域，是一本比较理想的PDE教材，美中不足的有两点，一是解的正则性理论涉及不多，二是没有涉及到流形上的PDE. 这方面内容比较全的是 Taylor巨著PDE（三卷本），只是内容太多了。不适合做教材用。这里推介Friedrich Sauvigny 的PDE（两卷本，Springer教材系列），篇幅上比Evans的PDE多了一百多百页，但远小于Taylor的PDE篇幅。很适合做教材，也适合自学用。
Friedrich Sauvigny - Partial Differential Equations 1: Foundations and Integral Representations (2nd edition)

Published: 2012-03-28 | ISBN: 1447129806 |462 p

This two-volume textbook provides comprehensive coverage of partial differential equations, spanning elliptic, parabolic, and hyperbolic types in two and several variables. In this first volume, special emphasis is placed on geometric and complex variable methods involving integral representations. The following topics are treated: • integration and differentiation on manifolds • foundations of functional analysis • Brouwer's mapping degree • generalized analytic functions • potential theory and spherical harmonics • linear partial differential equations This new second edition of this volume has been thoroughly revised and a new section on the boundary behavior of Cauchy’s integral has been added.


Friedrich Sauvigny - Partial Differential Equations 2: Functional Analytic Methods (2nd edition)

Published: 2012-03-30 | ISBN: 1447129830 |  469 p

This two-volume textbook provides comprehensive coverage of partial differential equations, spanning elliptic, parabolic, and hyperbolic types in two and several variables. In this second volume, special emphasis is placed on functional analytic methods and applications to differential geometry. The following topics are treated: solvability of operator equations in Banach spaces linear operators in Hilbert spaces and spectral theory Schauder's theory of linear elliptic differential equations weak solutions of differential equations nonlinear partial differential equations and characteristicsnonlinear elliptic systemsboundary value problems from differential geometryThis new second edition of this volume has been thoroughly revised and a new chapter on boundary value problems from differential geometry has been added. In the first volume, partial differential equations by integral representations are treated in a classical way. This textbook will be of particular use to graduate and postgraduate students interested in this field and will be of interest to advanced undergraduate students. It may also be used for independent study.