Lectures on elliptic and parabolic equations in Hölder spaces.

*(English)*Zbl 0865.35001
Graduate Studies in Mathematics. 12. Providence, RI: American Mathematical Society (AMS). xii, 164 p. (1996).

The present book is based on lecture notes of a two-quarter course on elliptic and parabolic equations which was given by the author in 1995 at the University of Minnesota. In the preface of the book, the author describes his aims in the following way: “My goal was not to try to cover as many subjects as possible but rather to concentrate on some basic facts and ideas of the modern theory of elliptic and parabolic equations and to lead the reader as far as possible in a short course. The presentation has been chosen in such a way that after having followed the book, the reader should acquire a good understanding of what kinds of results are available and what kind of technique is used to obtain them”. These two sentences really give a very good characterization of the book. It is short, but not condensed, well organized, and gives a stimulating presentation of basic aspects of the theory of elliptic and parabolic equations in Hölder spaces.

The book is designed as a textbook, therefore it cannot be expected that it contains new theoretical material, but nevertheless there are remarkable details: A priori estimates are obtained on the base of an idea of Safonov which makes use of potentials unnecessary, and the existence theorems are proved by using a method introduced by Browder. Since the Sobolev-space theory can be found in many textbooks, the present book is an interesting addition for students and instructors. In his preface, the author describes different ways how the presented material can be used for a course. It is nearly natural that a book of this type contains exercises, and indeed, one can find about 190 exercises. By way of contrast, the bibliography is very short, only 11 items are listed. I think that students would prefer a longer list. But this fact is not so important, it is more important to close this review with the following remark of the author: “My experience of following these notes at the University of Minnesota and in the Summer School at Cortona (Italy) in 1995 shows that students accept the subject very well and learn the material quickly and well”.

The book is designed as a textbook, therefore it cannot be expected that it contains new theoretical material, but nevertheless there are remarkable details: A priori estimates are obtained on the base of an idea of Safonov which makes use of potentials unnecessary, and the existence theorems are proved by using a method introduced by Browder. Since the Sobolev-space theory can be found in many textbooks, the present book is an interesting addition for students and instructors. In his preface, the author describes different ways how the presented material can be used for a course. It is nearly natural that a book of this type contains exercises, and indeed, one can find about 190 exercises. By way of contrast, the bibliography is very short, only 11 items are listed. I think that students would prefer a longer list. But this fact is not so important, it is more important to close this review with the following remark of the author: “My experience of following these notes at the University of Minnesota and in the Summer School at Cortona (Italy) in 1995 shows that students accept the subject very well and learn the material quickly and well”.

Reviewer: W.Watzlawek (Konstanz)