Linear and Quasilinear Elliptic Equations by Olga A. Ladyzhenskaya (PDF)

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⭐This book is a classic one in the theory of elliptic equations/systems of second order. The whole book is based on the Schauder theory for linear equations with smooth coefficients and the deeper theory of DeGiorgi on the boundedness and Hölderness of solutions to elliptic equations with merely bounded and measurable coefficients. Chapter 2 is devoted to discussing the so-called DeGiorgi classes and Chapter 3 deals with the Schauder theory. Discussions on the quasilinear equations with principal part in divergence form take place in the following chapter and the existence of solutions is built on Leray-Schauder’s fixed point theorem and the regularity results from Chapter 2 and Chapter 3. Chapter 5 consists of some basic facts about the variational problems, including finding the minima of variational integrals and their regularity properties. In Chapter 6, quasilinear equations of more general form is elaborated. Besides the DeGiorgi method, an idea of Bernstein is implemented to find an upper bound of the gradient of solutions. Questions regarding elliptic systems are also studied in Chapters 7 & 8, though not in their full generality. For more comprehensive treatment of elliptic systems, one may refer to Giaquinta’s book: “

⭐”. Some other devices, e.g. Moser’s method, of obtaining regularity properties of solutions to elliptic equations are also introduced at the end of the book. It should be mentioned that the authors have a companion volume on parabolic equations, namely

⭐. This book has almost all the above features of the current one under review. Lastly, if one wants to know more recent development on the theory of elliptic equations of second order, the book

⭐by Gilbarg and Trudinger should be a great reference.

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