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Second order elliptic equations and elliptic systems. Transl. from the Chinese by Bei Hu. (English) Zbl 0902.35003

Translations of Mathematical Monographs. 174. Providence, RI: American Mathematical Society (AMS). xiii, 246 p. (1998).
This book is based upon some lectures given by many well known experts in the area of elliptic PDE invited at the Nankai University in 1985. Later, all the material was suitably rearranged and was put in the form of a book with the aim to give a textbook for graduate students starting a Ph.D. program in PDE. The book contains many of the fundamental results in the area and, in this sense, it reaches the scope. It could be used for teaching PDE in graduate courses but not for first year students. In fact, elementary theory for the Laplacian is missing and so, the book is better understood by older students. However, let us now briefly describe its content.
The book is divided into two parts; the first one is devoted to second order elliptic equations while the other concerns elliptic systems. The book starts with a chapter concerning \(L^2\) theory for uniformly elliptic linear equations. After a detailed exposition of the Schauder theory, in the third chapter the reader finds the \(L^p\) theory for linear equations both in divergence and non-divergence form. A remark about chapter two is that Schauder estimates are derived in a way – introduced by Trudinger – which avoids potential theory. In Chapter four, De Giorgi-Moser-Nash results are proved concerning local and global Hölder continuity for weak solutions for linear equations. In Chapters 5, 6 and 7 quasilinear and fully nonlinear equations are briefly discussed. Regularity of generalized solutions is proved but - being the present book a textbook – only uniformly elliptic equations are discussed. The second half of the book concerns systems and starts, as usual, with some examples and remarks in order to explain to the reader the main differences between equations and systems. \(L^2\) theory is briefly discussed, Morrey and Campanato spaces are introduced and \(L^p\) theory is derived from \(L^2\) and \(BMO\) via interpolation. The last two chapters are devoted to existence and regularity for weak solutions of nonlinear elliptic systems. An appendix, very useful for the beginners, summarizes many notions and results used in the text.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J15 Second-order elliptic equations
35J45 Systems of elliptic equations, general (MSC2000)
35J60 Nonlinear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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