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Record Nr. |
UNINA9910814518403321 |
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Autore |
Lieberman Gary M. <1952-> |
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Titolo |
Oblique derivative problems for elliptic equations / / Gary M. Lieberman, Iowa State University, USA |
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Pubbl/distr/stampa |
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Singapore, : World Scientific, 2013 |
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New Jersey : , : World Scientific, , [2013] |
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�2013 |
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ISBN |
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Descrizione fisica |
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1 online resource (xv, 509 pages) : illustrations |
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Collana |
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Disciplina |
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Soggetti |
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Differential equations, Elliptic |
Mathematical physics |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Note generali |
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Description based upon print version of record. |
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Nota di bibliografia |
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Includes bibliographical references and index. |
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Nota di contenuto |
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Preface; Contents; 1. Pointwise Estimates; Introduction; 1.1 The maximum principle; 1.2 The definition of obliqueness; 1.3 The case c < 0, 0 0; 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 Holder 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; Exercises |
2. Classical Schauder Theory from a Modern PerspectiveIntroduction; 2.1 Definitions and properties of Holder spaces; 2.2 An alternative characterization of Holder 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; Notes; Exercises; 3. The Miller Barrier and Some Supersolutions for Oblique Derivative Problems; Introduction; 3.1 Theory of ordinary differential equations; 3.2 The Miller barrier construction |
3.3 Construction of supersolutions for Dirichlet data3.4 Construction of a supersolution for oblique derivative problems; 3.5 The strong maximum principle, revisited; 3.6 A Miller barrier for mixed boundary |
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value problems; Notes; Exercises; 4. Holder Estimates for First and Second Derivatives; Introduction; 4.1 C1, estimates for continuous; 4.2 Regularized distance; 4.3 Existence of solutions for continuous; 4.4 Holder gradient estimates for the Dirichlet problem; 4.5 C1, estimates with discontinuous in two dimensions; 4.6 C1, estimates for discontinuous in higher dimensions |
4.7 C2, estimatesNotes; Exercises; 5. Weak Solutions; Introduction; 5.1 Definitions and basic properties of weak derivatives; 5.2 Sobolev imbedding theorems; 5.3 Poincare'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 Holder spaces; 5.14 Schauder estimates; Notes; Exercises |
6. Strong SolutionsIntroduction; 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 Lp 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 Lp estimates for the second derivatives of strong solutions of differential equations; 6.8 Local Lp second derivative estimates near the boundary |
6.9 Existence of strong solutions for the oblique derivative problem |
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Sommario/riassunto |
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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. The author begins the book with basic results for linear oblique derivative problems and work through the theory for quasilinear and nonlinear problems. The final chapter discusses some of the applications. In addition, no |
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