Elliptic Partial Differential Equations (Courant Lecture Notes)

This volume is based on PDE courses given by the authors at the Courant Institute and at the University of Notre Dame (IN). Presented are basic methods for obtaining various a priori estimates for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. The equations considered in the book are linear, however, the presented methods also apply to nonlinear problems. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

Elliptic Partial Differential Equations by Qing Han and FangHua Lin is one of the best textbooks I know. It is the perfect introduction to PDE. In 150 pages or so it covers an amazing amount of wonderful and extraordinary useful material. I have used it as a textbook at both graduate and undergraduate levels which is possible since it only requires very little background material yet it covers an enormous amount of material. In my opinion it is a must read for all interested in analysis and geometry, and for all of my own PhD students it is indeed just that. I cannot say enough good things about it—it is a wonderful book. —Tobias Colding This volume is based on PDE courses given by the authors at the Courant Institute and at the University of Notre Dame, Indiana. Presented are basic methods for obtaining various a priori estimates for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. The equations considered in the book are linear; however, the presented methods also apply to nonlinear problems. This second edition has been thoroughly revised and in a new chapter the authors discuss several methods for proving the existence of solutions of primarily the Dirichlet problem for various types of elliptic equations.

Cover......Page 1 Title page......Page 2 Contents......Page 6 Preface......Page 8 1.2. Mean Value Properties......Page 10 1.3. Fundamental Solutions......Page 17 1.4. Maximum Principles......Page 24 1.5. Energy Method......Page 28 2.2. Strong Maximum Principle......Page 34 2.3. A Priori Estimates......Page 39 2.4. Gradient Estimates......Page 42 2.5. Alexandroff Maximum Principle......Page 46 2.6. Moving Plane Method......Page 52 3.1. Guide......Page 56 3.2. Growth of Local Integrals......Page 57 3.3. Hölder Continuity of Solutions......Page 64 3.4. Hölder Continuity of Gradients......Page 69 4.2. Local Boundedness......Page 76 4.3. Hölder Continuity......Page 87 4.4. Moser's Harnack Inequality......Page 92 4.5. Nonlinear Equations......Page 102 5.2. Alexandroff Maximum Principle......Page 108 5.3. Harnack Inequality......Page 113 5.4. Schauder Estimates......Page 122 5.5. W... Estimates......Page 126 Bibliography......Page 132