Elliptic Equations in Polyhedral Domains

Elliptic Equations in Polyhedral Domains PDF

Author: V. G. Maz_i_a

Publisher: American Mathematical Soc.

Published: 2010-04-22

Total Pages: 618

ISBN-13: 0821849832

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This is the first monograph which systematically treats elliptic boundary value problems in domains of polyhedral type. The authors mainly describe their own recent results focusing on the Dirichlet problem for linear strongly elliptic systems of arbitrary order, Neumann and mixed boundary value problems for second order systems, and on boundary value problems for the stationary Stokes and Navier-Stokes systems. A feature of the book is the systematic use of Green's matrices. Using estimates for the elements of these matrices, the authors obtain solvability and regularity theorems for the solutions in weighted and non-weighted Sobolev and Holder spaces. Some classical problems of mathematical physics (Laplace and biharmonic equations, Lame system) are considered as examples. Furthermore, the book contains maximum modulus estimates for the solutions and their derivatives. The exposition is self-contained, and an introductory chapter provides background material on the theory of elliptic boundary value problems in domains with smooth boundaries and in domains with conical points. The book is destined for graduate students and researchers working in elliptic partial differential equations and applications.

Elliptic Boundary Value Problems on Corner Domains

Elliptic Boundary Value Problems on Corner Domains PDF

Author: Monique Dauge

Publisher: Springer

Published: 2006-11-14

Total Pages: 266

ISBN-13: 3540459421

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This research monograph focusses on a large class of variational elliptic problems with mixed boundary conditions on domains with various corner singularities, edges, polyhedral vertices, cracks, slits. In a natural functional framework (ordinary Sobolev Hilbert spaces) Fredholm and semi-Fredholm properties of induced operators are completely characterized. By specially choosing the classes of operators and domains and the functional spaces used, precise and general results may be obtained on the smoothness and asymptotics of solutions. A new type of characteristic condition is introduced which involves the spectrum of associated operator pencils and some ideals of polynomials satisfying some boundary conditions on cones. The methods involve many perturbation arguments and a new use of Mellin transform. Basic knowledge about BVP on smooth domains in Sobolev spaces is the main prerequisite to the understanding of this book. Readers interested in the general theory of corner domains will find here a new basic theory (new approaches and results) as well as a synthesis of many already known results; those who need regularity conditions and descriptions of singularities for numerical analysis will find precise statements and also a means to obtain further one in many explicit situtations.

Graded Finite Element Methods for Elliptic Problems in Nonsmooth Domains

Graded Finite Element Methods for Elliptic Problems in Nonsmooth Domains PDF

Author: Hengguang Li

Publisher: Springer Nature

Published: 2022-09-01

Total Pages: 186

ISBN-13: 3031058216

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This book develops a class of graded finite element methods to solve singular elliptic boundary value problems in two- and three-dimensional domains. It provides an approachable and self-contained presentation of the topic, including both the mathematical theory and numerical tools necessary to address the major challenges imposed by the singular solution. Moreover, by focusing upon second-order equations with constant coefficients, it manages to derive explicit results that are accessible to the broader computation community. Although written with mathematics graduate students and researchers in mind, this book is also relevant to applied and computational mathematicians, scientists, and engineers in numerical methods who may encounter singular problems.

Constructive Methods for Elliptic Equations

Constructive Methods for Elliptic Equations PDF

Author: R.P. Gilbert

Publisher: Springer

Published: 2006-11-15

Total Pages: 405

ISBN-13: 3540379533

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Primarily these lectures are a report on recent work by the Indiana University group working on Function theoretic methods as applied to the theory of partial differential equations.

The Numerical Solution of Elliptic Equations

The Numerical Solution of Elliptic Equations PDF

Author: Garrett Birkhoff

Publisher: SIAM

Published: 1971-01-01

Total Pages: 100

ISBN-13: 9780898710014

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A concise survey of the current state of knowledge in 1972 about solving elliptic boundary-value eigenvalue problems with the help of a computer. This volume provides a case study in scientific computing?the art of utilizing physical intuition, mathematical theorems and algorithms, and modern computer technology to construct and explore realistic models of problems arising in the natural sciences and engineering.

Elliptic, Hyperbolic and Mixed Complex Equations with Parabolic Degeneracy

Elliptic, Hyperbolic and Mixed Complex Equations with Parabolic Degeneracy PDF

Author: Guo Chun Wen

Publisher: World Scientific

Published: 2008

Total Pages: 453

ISBN-13: 9812779434

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In the recent half-century, many mathematicians have investigated various problems on several equations of mixed type and obtained interesting results, with important applications to gas dynamics. However, the Tricomi problem of general mixed type equations of second order with parabolic degeneracy has not been completely solved, particularly the Tricomi and Frankl problems for general Chaplygin equation in multiply connected domains posed by L Bers, and the existence, regularity of solutions of the above problems for mixed equations with non-smooth degenerate curve in several domains posed by J M Rassias. The method revealed in this book is unlike any other, in which the hyperbolic number and hyperbolic complex function in hyperbolic domains, and the complex number and complex function in elliptic domains are used. The corresponding problems for first order complex equations with singular coefficients are first discussed, and then the problems for second order complex equations are considered, where we pose the new partial derivative notations and complex analytic methods such that the forms of the above first order complex equations in hyperbolic and elliptic domains are wholly identical. In the meantime, the estimates of solutions for the above problems are obtained, hence many open problems including the above TricomiOCo Bers and TricomiOCoFranklOCoRassias problems can be solved. Sample Chapter(s). Chapter 1: Elliptic Complex Equations of First Order (247 KB). Contents: Elliptic Complex Equations of First Order; Elliptic Complex Equations of Second Order; Hyperbolic Complex Equations of First and Second Orders; First Order Complex Equations of Mixed Type; Second Order Linear Equations of Mixed Type; Second Order Quasilinear Equations of Mixed Type. Readership: Graduate students and academics in analysis, differential equations and applied mathematics.

Oblique Derivative Problems For Elliptic Equations

Oblique Derivative Problems For Elliptic Equations PDF

Author: Gary M Lieberman

Publisher: World Scientific

Published: 2013-03-26

Total Pages: 526

ISBN-13: 9814452343

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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, notes to each chapter give a history of the topics in that chapter and suggestions for further reading.

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