Fine regularity of solutions of elliptic partial differential equations / Jan Malý, William P. Ziemer.

Format:

Book

Author/Creator:

Malý, Jan, author.

Ziemer, William P., author.

Series:

Mathematical surveys and monographs ; no. 51.

Mathematical surveys and monographs, 0076-5376 ; volume 51

Language:

English

Subjects (All):

Potential theory (Mathematics).

Differential equations, Elliptic--Numerical solutions.

Differential equations, Elliptic.

Boundary value problems.

Physical Description:

1 online resource (309 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1997]

Language Note:

English

Summary:

The primary objective of this book is to give a comprehensive exposition of results surrounding the work of the authors concerning boundary regularity of weak solutions of second order elliptic quasilinear equations in divergence form. The book also contains a complete development of regularity of solutions ofvariational inequalities, including the double obstacle problem, where the obstacles are allowed to be discontinuous. The book concludes with a chapter devoted to the existence theory thus providing the reader with a complete treatment of the subject ranging from regularity of weak solutions to the existence of weak solutions.

Contents:

""Contents""; ""Preface""; ""Basic Notation""; ""Chapter 1. Preliminaries""; ""1.1 Basic results""; ""1.1.1 Covering theorems""; ""1.1.2 Densities of measures""; ""1.1.3 The maximal function and its applications""; ""1.2 Potential estimates""; ""1.3 Sobolev spaces""; ""1.3.1 Inequalities""; ""1.3.2 Imbeddings""; ""1.3.3 Pointwise differentiability of Sobolev functions""; ""1.3.4 Spaces Y[sup(1,p)]""; ""1.3.5 Adams' inequality""; ""1.3.6 Bessel and Riesz potentials""; ""1.4 Historical notes""; ""Chapter 2. Potential Theory""; ""2.1 Capacity""; ""2.1.1 Comparison of capacities

capacities of balls""""2.1.2 Polar sets""; ""2.1.3 Quasicontinuity""; ""2.1.4 Multipliers""; ""2.1.5 Capacity and energy minimizers""; ""2.1.6 Thinness""; ""2.1.7 Capacity and Hausdorff measure""; ""2.1.8 Lebesgue points for Sobolev functions""; ""2.2 Laplace equation""; ""2.2.1 Green potentials""; ""2.2.2 Classical thinness""; ""2.2.3 Dirichlet problem and the Wiener criterion""; ""2.3 Regularity of minimizers""; ""2.3.1 Abstract minimization""; ""2.3.2 Minimizers and weak solutions""; ""2.3.3 Higher regularity""; ""2.3.4 The De Giorgi method""; ""2.3.5 Moser's iteration technique""

""2.3.6 Removable singularities""""2.3.7 Estimates of supersolutions""; ""2.3.8 Estimates of energy minimizers""; ""2.3.9 Dirichlet problem""; ""2.3.10 Application of thinness: the Wiener criterion""; ""2.4 Fine topology""; ""2.5 Fine Sobolev spaces""; ""2.6 Historical notes""; ""Chapter 3. Quasilinear Equations""; ""3.1 Basic properties of weak solutions""; ""3.1.1 Upper bounds for weak solutions""; ""3.1.2 Weak Harnack inequality""; ""3.1.3 Removable sets for weak solutions""; ""3.2 Higher regularity of equations with differentiate structure""; ""3.3 Historical notes""

""Chapter 4. Fine Regularity Theory""""4.1 Basic energy estimates""; ""4.2 Sufficiency of the Wiener condition for boundary regularity""; ""4.2.1 The special case of harmonic functions""; ""4.3 Necessity of the Wiener condition for boundary regularity""; ""4.3.1 Main estimate""; ""4.3.2 Necessity of the Wiener condition""; ""4.4 Equations with measure data""; ""4.5 Historical notes""; ""Chapter 5. Variational Inequalities � Regularity""; ""5.1 Differential operators with measurable coefficients""; ""5.1.1 Continuity in the presence of irregular obstacles""

""5.1.2 The modulus of continuity""""5.2 Differential operators with differentiate structure""; ""5.3 Historical notes""; ""Chapter 6. Existence Theory""; ""6.1 Existence of solutions to variational inequalities""; ""6.1.1 Pseudomonotone operators""; ""6.1.2 Variational problems existence of bounded solutions ""; ""6.1.3 Variational problems leading to unbounded solutions ""; ""6.2 The Dirichlet problem for equations with differentiable structure""; ""6.3 Historical notes""; ""References""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""J""; ""K""; ""L""; ""M""

""N""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 273-281) and indexes.

Description based on print version record.

ISBN:

1-4704-1282-9