Oblique derivative problems for elliptic equations

• Verfasst von: Lieberman, Gary M. [VerfasserIn]
• Titel: Oblique derivative problems for elliptic equations
• Verf.angabe: Gary M Lieberman, Iowa State University, USA
• Verlagsort: New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai
• Verlag: World Scientific
• E-Jahr: 2013
• Jahr: [2013]
• Umfang: 1 Online-Ressource (xv, 509 Seiten)
• Illustrationen: Diagramme
• Fussnoten: Includes bibliographical references (p. 493-505) and index
• ISBN: 978-981-4452-33-5
• DOI: doi:10.1142/8679
• Schlagwörter: (s)Elliptische Differentialgleichung
• Datenträger: Online-Ressource
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausg.: Lieberman, Gary M.: Oblique derivative problems for elliptic equations. - New Jersey [u.a.] : World Scientific Publ., 2013. - XV, 509 S.
• RVK-Notation: SK 560
• K10plus-PPN: 1690509082
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

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 supersolution for oblique derivative problems. 3.5. The strong maximum principle, revisited. 3.6. A Miller barrier for mixed boundary value problems -- 4. Hölder estimates for first and second derivatives. 4.1. C[symbol] estimates for continuous [symbol]. 4.2. Regularized distance. 4.3. Existence of solutions for continuous [symbol]. 4.4. Hölder gradient estimates for the Dirichlet problem. 4.5. C[symbol] estimates with discontinuous [symbol] in two dimensions. 4.6. C[symbol] estimates for discontinuous [symbol] in higher dimensions. 4.7. C[symbol] estimates -- 5. Weak solutions. 5.1. Definitions and basic properties of weak derivatives. 5.2. Sobolev imbedding theorems. 5.3. Poincaré's inequality. 5.4. The weak maximum principle. 5.5. Trace theorems. 5.6. Existence of weak solutions. 5.7. Higher regularity of solutions. 5.8. Global boundedness of weak solutions. 5.9. The local maximum principle. 5.10. The DeGiorgi class. 5.11. Membership of supersolutions in the De Giorgi class. 5.12. Consequences of the local estimates. 5.13. Integral characterizations of Hölder spaces. 5.14. Schauder estimates -- 6. Strong solutions. 6.1. Pointwise estimates for strong solutions. 6.2. A sharp trace theorem. 6.3. Results from harmonic analysis. 6.4. Some further estimates for boundary value problems in a spherical cap. 6.5. L[symbol] estimates for solutions of constant coefficient problems in a spherical cap. 6.6. Local estimates for strong solutions of constant coefficient problems. 6.7. Local interior L[symbol] estimates for the second derivatives of strong solutions of differential equations. 6.8. Local L[symbol] second derivative estimates near the boundary. 6.9. Existence of strong solutions for the oblique derivative problem.

Abstract

7. Viscosity solutions of oblique derivative problems. 7.1. Definitions and notation. 7.2. The Theorem of Aleksandrov. 7.3. Preliminary results for the comparison theorem for viscosity solutions. 7.4. The comparison principle for viscosity sub- and supersolutions. 7.5. A test function construction for the oblique derivative problem. 7.6. The comparison principle for oblique derivative problems. 7.7. Existence and uniqueness of viscosity solutions -- 8. Pointwise bounds for solutions of problems with quasilinear equations. 8.1. Maximum estimates for nondivergence equations. 8.2. Hölder estimates for nondivergence equations. 8.3. Maximum estimates for conormal problems. 8.4. Hölder estimates for conormal problems -- 9. Gradient estimates for general form oblique derivative problems. 9.1. Interior gradient bounds. 9.2. A simple boundary value problem. 9.3. Gradient estimates for general boundary conditions. General considerations. 9.4. Global gradient estimates for general boundary conditions and false mean curvature equations I. 9.5. Global gradient estimates for general boundary conditions and false mean curvature equations II. 9.6. Local gradient estimates. 9.7. Gradient estimates for capillary-type problems -- 10. Gradient estimates for the conormal derivative problems. 10.1. The Sobolev inequality of Michael and Simon. 10.2. The interior gradient bound. 10.3. Preliminaries for estimates. 10.4. Gradient bounds for the conormal problem -- 11. Higher order estimates and existence of solutions for quasilinear oblique derivative problems. 11.1. The Hölder gradient estimate for conormal problems. 11.2. A solvability theorem. 11.3. Existence results and estimates for linear equations and nonlinear boundary conditions in spherical caps. 11.4. Estimates and existence results for linear equations and nonlinear boundary conditions in general domains. 11.5. Mixed boundary value problems for simple quasilinear differential equations and nonlinear boundary conditions in spherical caps. 11.6. Hölder gradient estimates for quasilinear equations. 11.7. A basic existence theorem for quasilinear elliptic equations with nonlinear boundary conditions. 11.8. Second derivative Hölder estimates. 11.9. Existence theorems for our examples -- 12. Oblique derivative problems for fully nonlinear elliptic equations. 12.1. Maximum estimates, comparison principles, and a uniqueness theorem. 12.2. Second derivative Hölder estimates. 12.3. Second derivative Hölder estimates for solutions of oblique derivative problems. 12.4. Uniformly elliptic fully nonlinear problems.

Abstract

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