Partial Differential Equations of Mathematical Physics (Dover Books on Physics)

Prerequisite knowledge of ODEs and intro PDEs needed.Also some potential theory.It's an excellent book. and agrees w/ other books on PDEs an Potential Theory.Recommend (all on Dover) simultaneous study of:1) Potential Theory, O. Kellog2) Intro to PDEs w/ Applications, E.C. Zachmanoglou3) Intro to ODEs, E. CoddingtonWill be continuing my study of this book.Mr. Sobolev has very interesting perspectives on this PDEs subject.

Quite frankly, I had almost forgotten about this book. However, while working my way through Gabriel Barton's Book, Elements Of Green's Functions And Propagation, then his references, I was struck by what he wrote regarding this text: "accessibly written...a truly exceptional text, head and shoulders above most others with similar titles." Thus, I retrieved the book from my bookshelf. I confess to having ignored it for some time. The reason is obvious, as it is decidedly much more advanced than others "with similar titles."(1) Sobolev is advanced, this implies that a course in analysis has already been assimilated. Happily, if you have gone through much of Weinberger's text, A First Course in Partial Differential Equations, Sobolev will be easier to assail. Onward, then:(2) Of particular note is chapter six: multiple integrals and lebesgue integration (this is a "measure theory first," introduction). Also, if a sentence such as "any closed set can always be taken to be the intersection of a nest of open sets" is unfamiliar, this will be a tough chapter. On the positive side, the exposition is exceptionally lucid, and the initial five chapters (referred to by Sobolev as "lectures") are, as a whole, easier than the sixth lecture.(3) Surprisingly few intermediate steps are left for the reader. Two examples: glance at pages,134 and 135. Here you find the computations pedestrian and spelled-out for the reader (this is lecture eight, heat conduction). If you have glanced at Weinberger, then lectures eight and nine will not pose too much of an issue.(4) You meet interesting approaches in Lecture fifteen, properties of potentials. A glance at this chapter affords an opportunity to observe interplay between inequalities and differential geometric analysis. A most interesting analysis.(5) Next up, integral equations. Read: "the properties of potentials which we established in the last lecture enable us to solve the Dirichlet and Neumann problems by reducing the problems to the form of integral equations." (page 225). Lectures 18 and 19 revisit this theme. That is, lecture 18 will present theory, lecture 19 presents applications.(6) Green's Functions, next: "we will begin with a very simple case." (page 265). Green will occupy two chapters.These two chapters are ancillary to Barton's textbook.(7) Lecture twenty-two: The lecture has a major theme, inequalities: Of Minkowski and Schwarz. A fine exposition.(8) Fourier and separation-of-variables. We read: "the reader who has mastered the arguments set out in the previous lectures will have no difficulty in understanding immediately how and in what circumstances the Fourier method enables the solution of a problem to be found." (page 327).(9) Next, examining more general integral equations. So you will for three chapters following. After that, more Fourier methods, culminating in an exposition of spherical functions and harmonics--the final two lectures.(10) Concluding: Given adequate preparation (a first course in analysis) Sobolev is a fine text (no student exercises).In conjunction with Barton's text (with its copious supply of student exercises) a sound foundation is achieved.The exposition is exceptional, most intermediate steps are made explicit, the analysis (the proofs) fascinating.As preliminary, read Sobolev's fine forty-five page exposition of partial differential equations in Mathematics:Its Content, Methods and Meaning (volume two, MIT Press).Highly recommended for advanced students.