Spectral Problems Associated With Corner Singularities Of Solutions To Elliptic Equations
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Spectral Problems Associated With Corner Singularities Of Solutions To Elliptic Equations
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Author : Vladimir Kozlov
language : en
Publisher: American Mathematical Soc.
Release Date : 2001
Spectral Problems Associated With Corner Singularities Of Solutions To Elliptic Equations written by Vladimir Kozlov and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.
This book focuses on the analysis of eigenvalues and eigenfunctions that describe singularities of solutions to elliptic boundary value problems in domains with corners and edges. The authors treat both classical problems of mathematical physics and general elliptic boundary value problems. The volume is divided into two parts: The first is devoted to the power-logarithmic singularities of solutions to classical boundary value problems of mathematical physics. The second deals with similar singularities for higher order elliptic equations and systems. Chapter 1 collects basic facts concerning operator pencils acting in a pair of Hilbert spaces. Related properties of ordinary differential equations with constant operator coefficients are discussed and connections with the theory of general elliptic boundary value problems in domains with conic vertices are outlined. New results are presented. Chapter 2 treats the Laplace operator as a starting point and a model for the subsequent study of angular and conic singularities of solutions. Chapter 3 considers the Dirichlet boundary condition beginning with the plane case and turning to the space problems. Chapter 4 investigates some mixed boundary conditions. The Stokes system is discussed in Chapters 5 and 6, and Chapter 7 concludes with the Dirichlet problem for the polyharmonic operator. Chapter 8 studies the Dirichlet problem for general elliptic differential equations of order 2m in an angle. In Chapter 9, an asymptotic formula for the distribution of eigenvalues of operator pencils corresponding to general elliptic boundary value problems in an angle is obtained. Chapters 10 and 11 discuss the Dirichlet problem for elliptic systems of differential equations of order 2 in an n-dimensional cone. Chapter 12 studies the Neumann problem for general elliptic systems, in particular with eigenvalues of the corresponding operator pencil in the strip $\mid {\Re} \lambda - m + /2n \mid \leq 1/2$. It is shown that only integer numbers contained in this strip are eigenvalues. Applications are placed within chapter introductions and as special sections at the end of chapters. Prerequisites include standard PDE and functional analysis courses.
Spectral Problems Associated With Corner Singularities Of Solutions To Elliptic Equations
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Author : Vladimir Kozlov
language : en
Publisher: American Mathematical Society(RI)
Release Date : 2014-06-06
Spectral Problems Associated With Corner Singularities Of Solutions To Elliptic Equations written by Vladimir Kozlov and has been published by American Mathematical Society(RI) this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-06-06 with MATHEMATICS categories.
This text focuses on the analysis of eigenvalues and eigenfunctions that describe singularities of solutions to elliptic boundary value problems in domains with corners and edges. The authors treat both classical problems of mathematical physics and general elliptic boundary value problems. The volume is divided into two parts: The first is devoted to the power-logarithmic singularities of solutions to classical boundary value problems of mathematical physics. The second deals with similar singularities for higher order elliptic equations and systems. Chapter 1 collects basic facts concerning operator pencils acting in a pair of Hilbert spaces. Related properties of ordinary differential equations with constant operator coefficients are discussed and connections with the theory of general elliptic boundary value problems in domains with conic vertices are outlined. New results are presented. Chapter 2 treats the Laplace operator as a starting point and a model for the subsequent study of angular and conic singularities of solutions. Chapter 3 considers the Dirichlet boundary condition beginning with the plane case and turning to the space problems.
Finite Element Error Analysis For Pde Constrained Optimal Control Problems
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Author : Dieter Sirch
language : en
Publisher: Logos Verlag Berlin GmbH
Release Date : 2010
Finite Element Error Analysis For Pde Constrained Optimal Control Problems written by Dieter Sirch and has been published by Logos Verlag Berlin GmbH this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Mathematics categories.
Subject of this work is the analysis of numerical methods for the solution of optimal control problems governed by elliptic partial differential equations. Such problems arise, if one does not only want to simulate technical or physical processes but also wants to optimize them with the help of one or more influence variables. In many practical applications these influence variables, so called controls, cannot be chosen arbitrarily, but have to fulfill certain inequality constraints. The numerical treatment of such control constrained optimal control problems requires a discretization of the underlying infinite dimensional function spaces. To guarantee the quality of the numerical solution one has to estimate and to quantify the resulting approximation errors. In this thesis a priori error estimates for finite element discretizations are proved in case of corners or edges in the underlying domain and nonsmooth coefficients in the partial differential equation. These facts influence the regularity properties of the solution and require adapted meshes to get optimal convergence rates. Isotropic and anisotropic refinement strategies are given and error estimates in polygonal and prismatic domains are proved. The theoretical results are confirmed by numerical tests.
Elliptic Theory On Singular Manifolds
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Author : Vladimir E. Nazaikinskii
language : en
Publisher: CRC Press
Release Date : 2005-08-12
Elliptic Theory On Singular Manifolds written by Vladimir E. Nazaikinskii and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-08-12 with Mathematics categories.
The analysis and topology of elliptic operators on manifolds with singularities are much more complicated than in the smooth case and require completely new mathematical notions and theories. While there has recently been much progress in the field, many of these results have remained scattered in journals and preprints. Starting from an ele
Multi Layer Potentials And Boundary Problems
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Author : Irina Mitrea
language : en
Publisher: Springer
Release Date : 2013-01-05
Multi Layer Potentials And Boundary Problems written by Irina Mitrea and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-01-05 with Mathematics categories.
Many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator in a given domain. When the differential operator under discussion is of second order a variety of tools are available for dealing with such problems, including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. When the differential operator is of higher-order (as is the case, e.g., with anisotropic plate bending when one deals with a fourth order operator) only a few options could be successfully implemented. In the 1970s Alberto Calderón, one of the founders of the modern theory of Singular Integral Operators, advocated the use of layer potentials for the treatment of higher-order elliptic boundary value problems. The present monograph represents the first systematic treatment based on this approach. This research monograph lays, for the first time, the mathematical foundation aimed at solving boundary value problems for higher-order elliptic operators in non-smooth domains using the layer potential method and addresses a comprehensive range of topics, dealing with elliptic boundary value problems in non-smooth domains including layer potentials, jump relations, non-tangential maximal function estimates, multi-traces and extensions, boundary value problems with data in Whitney–Lebesque spaces, Whitney–Besov spaces, Whitney–Sobolev- based Lebesgue spaces, Whitney–Triebel–Lizorkin spaces,Whitney–Sobolev-based Hardy spaces, Whitney–BMO and Whitney–VMO spaces.
Analysis Partial Differential Equations And Applications
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Author : Alberto Cialdea
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-01-14
Analysis Partial Differential Equations And Applications written by Alberto Cialdea and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-01-14 with Mathematics categories.
This volume includes several invited lectures given at the International Workshop "Analysis, Partial Differential Equations and Applications", held at the Mathematical Department of Sapienza University of Rome, on the occasion of the 70th birthday of Vladimir G. Maz'ya, a renowned mathematician and one of the main experts in the field of pure and applied analysis. The book aims at spreading the seminal ideas of Maz'ya to a larger audience in faculties of sciences and engineering. In fact, all articles were inspired by previous works of Maz'ya in several frameworks, including classical and contemporary problems connected with boundary and initial value problems for elliptic, hyperbolic and parabolic operators, Schrödinger-type equations, mathematical theory of elasticity, potential theory, capacity, singular integral operators, p-Laplacians, functional analysis, and approximation theory. Maz'ya is author of more than 450 papers and 20 books. In his long career he obtained many astonishing and frequently cited results in the theory of harmonic potentials on non-smooth domains, potential and capacity theories, spaces of functions with bounded variation, maximum principle for higher-order elliptic equations, Sobolev multipliers, approximate approximations, etc. The topics included in this volume will be particularly useful to all researchers who are interested in achieving a deeper understanding of the large expertise of Vladimir Maz'ya.
Spherical Harmonics And Approximations On The Unit Sphere An Introduction
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Author : Kendall Atkinson
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-02-17
Spherical Harmonics And Approximations On The Unit Sphere An Introduction written by Kendall Atkinson and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-02-17 with Mathematics categories.
These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. The notes are intended for graduate students in the mathematical sciences and researchers who are interested in solving problems involving partial differential and integral equations on the unit sphere, especially on the unit sphere in three-dimensional Euclidean space. Some related work for approximation on the unit disk in the plane is also briefly discussed, with results being generalizable to the unit ball in more dimensions.
Around The Research Of Vladimir Maz Ya Ii
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Author : Ari Laptev
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-12-05
Around The Research Of Vladimir Maz Ya Ii written by Ari Laptev and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-12-05 with Mathematics categories.
Topics of this volume are close to scientific interests of Professor Maz'ya and use, directly or indirectly, the fundamental influential Maz'ya's works penetrating, in a sense, the theory of PDEs. In particular, recent advantages in the study of semilinear elliptic equations, stationary Navier-Stokes equations, the Stokes system in convex polyhedra, periodic scattering problems, problems with perturbed boundary at a conic point, singular perturbations arising in elliptic shells and other important problems in mathematical physics are presented.
Arithmetic Differential Equations
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Author : Alexandru Buium
language : en
Publisher: American Mathematical Soc.
Release Date : 2005
Arithmetic Differential Equations written by Alexandru Buium and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.
For most of the book the only prerequisites are the basic facts of algebraic geometry and number theory."--BOOK JACKET.
Boundary Element Methods
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Author : Stefan A. Sauter
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-11-01
Boundary Element Methods written by Stefan A. Sauter and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-01 with Mathematics categories.
This work presents a thorough treatment of boundary element methods (BEM) for solving strongly elliptic boundary integral equations obtained from boundary reduction of elliptic boundary value problems in $\mathbb{R}^3$. The book is self-contained, the prerequisites on elliptic partial differential and integral equations being presented in Chapters 2 and 3. The main focus is on the development, analysis, and implementation of Galerkin boundary element methods, which is one of the most flexible and robust numerical discretization methods for integral equations. For the efficient realization of the Galerkin BEM, it is essential to replace time-consuming steps in the numerical solution process with fast algorithms. In Chapters 5-9 these methods are developed, analyzed, and formulated in an algorithmic way.