Oblique derivative problems for elliptic equations

This book gives an up-to-date exposition on the theory of oblique derivative problems for elliptic equations. The modern analysis of shock reflection was made possible by the theory of oblique derivative problems developed by the author. Such problems also arise in many other physical situations such as the shape of a capillary surface and problems of optimal transportation. We begin with basic results for linear oblique derivative problems and work through the theory for quasilinear and nonlinear problems. A final chapter discusses some of the applications. In addition, notes to each chapter give a history of the topics in that chapter and suggestions for further reading.

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1. Pointwise estimates. 1.1. The maximum principle. 1.2. The definition of obliqueness. 1.3. The case c <0, [symbol]. 1.4. A generalized change of variables formula. 1.5. The Aleksandrov-Bakel'man-Pucci maximum principles. 1.6. The interior weak Harnack inequality. 1.7. The weak Harnack inequality at the boundary. 1.8. The strong maximum principle and uniqueness. 1.9. Hölder continuity. 1.10. The local maximum principle. 1.11. Pointwise estimates for solutions of mixed boundary value problems. 1.12. Derivative bounds for solutions of elliptic equations -- 2. Classical Schauder theory from a modern perspective. 2.1. Definitions and properties of Hölder spaces. 2.2. An alternative characterization of Hölder spaces. 2.3. An existence result. 2.4. Basic interior estimates. 2.5. The Perron process for the Dirichlet problem. 2.6. A model mixed boundary value problem. 2.7. Domains with curved boundary. 2.8. Fredholm-Riesz-Schauder theory -- 3. The Miller barrier and some supersolutions for oblique derivative problems. 3.1. Theory of ordinary differential equations. 3.2. The Miller barrier construction. 3.3. Construction of supersolutions for Dirichlet data. 3.4. Construction of a superso…