Asymptotic theory of dynamic boundary value problems in irregular domains /

Asymptotic theory of dynamic boundary value problems in irregular domains /

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Bibliographic Details
Author / Creator:Korikov, Dmitrii.
Imprint:Cham : Birkhäuser, 2021.
Description:1 online resource (404 p.).
Language:English
Series:Operator Theory, Advances and Applications ; v. 284
Operator theory, advances and applications ; v. 284.
Subject:
Boundary value problems -- Asymptotic theory.
Boundary value problems -- Asymptotic theory.
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/12611909
Hidden Bibliographic Details
Other authors / contributors:Plamenevskiĭ, B. A.
Sarafanov, Oleg.
ISBN:9783030653729
3030653722
9783030653712
3030653714
Notes:Description based upon print version of record.
4.6 Boundary Value Problem in a Cone in a Scaleof Weighted Spaces.
Includes bibliographical references.
Summary:This book considers dynamic boundary value problems in domains with singularities of two types. The first type consists of "edges" of various dimensions on the boundary; in particular, polygons, cones, lenses, polyhedra are domains of this type. Singularities of the second type are "singularly perturbed edges" such as smoothed corners and edges and small holes. A domain with singularities of such type depends on a small parameter, whereas the boundary of the limit domain (as the parameter tends to zero) has usual edges, i.e. singularities of the first type. In the transition from the limit domain to the perturbed one, the boundary near a conical point or an edge becomes smooth, isolated singular points become small cavities, and so on. In an "irregular" domain with such singularities, problems of elastodynamics, electrodynamics and some other dynamic problems are discussed. The purpose is to describe the asymptotics of solutions near singularities of the boundary. The presented results and methods have a wide range of applications in mathematical physics and engineering. The book is addressed to specialists in mathematical physics, partial differential equations, and asymptotic methods.
Other form:Print version: Korikov, Dmitrii Asymptotic Theory of Dynamic Boundary Value Problems in Irregular Domains Cham : Springer International Publishing AG,c2021 9783030653712
Standard no.:10.1007/978-3-030-65372-9
Table of Contents:
  • Intro
  • Preface
  • Contents
  • 1 Introduction
  • 2 Wave Equation in Domains with Edges
  • 2.1 Dirichlet Problem for the Wave Equation
  • 2.1.1 Function Spaces in a Wedge and in a Cone
  • 2.1.2 Problem in a Wedge: Problem with Parameter in a Cone: Existence of Solutions
  • 2.1.3 Weighted Combined Estimates
  • 2.1.4 Operators in the Scale of Weighted Spaces
  • 2.1.5 Asymptotics of Solutions Near the Vertex of a Cone or Near the Edge of a Wedge
  • 2.1.6 Explicit Formulas for the Coefficients in Asymptotics
  • 2.1.7 Problem in a Bounded Domain with Conical Points
  • 2.1.8 Problem in a Bounded Domain: Asymptotics of Solutions Near an Internal Point
  • 2.2 Neumann Problem for the Wave Equation
  • 2.2.1 Statement of the Problem: Preliminaries
  • 2.2.2 Weighted Combined Estimates for Solutions to Problem (2.138), (2.139)
  • 2.2.3 Operator of the Boundary Value Problem in a Cone
  • 2.2.4 Boundary Value Problem in a Cone in the Scale of Weighted Spaces
  • 2.2.5 Asymptotic Expansions of Solutions to the Problem in a Cone
  • 2.2.6 Problem in a Wedge
  • 2.2.7 Explicit Formulas for the Coefficients in Asymptotics
  • 2.2.8 Problem in a Bounded Domain with Conical Points
  • 3 Hyperbolic Systems in Domains with Conical Points
  • 3.1 Cauchy-Dirichlet Problem
  • 3.1.1 Combined Estimate for Solutions of the Problem in a Cone
  • 3.1.2 Operator of the Boundary Value Problem in a Cone: The Existence and Uniqueness of Solutions
  • 3.1.3 The Boundary Value Problem in a Cone in the Scale of Weighted Spaces
  • 3.1.4 Asymptotics of Solutions of the Problem in a Cone
  • 3.1.5 The Problem in a Wedge
  • 3.2 Neumann Problem
  • 3.2.1 The Model Problems in a Cone: A Strong Solution
  • 3.2.2 Weighted Estimates of Solutions of the Problem with Parameter in a Cone
  • 3.2.3 The Problem with Parameter in a Cone: A Scale of Weighted Spaces
  • 3.2.4 The Asymptotics of Solutions
  • 3.2.5 A Bounded Domain with a Conical Point
  • 4 Elastodynamics in Domains with Edges
  • 4.1 Introduction
  • 4.2 Homogeneous Energy Estimates on Solutions of Boundary Value Problems with Parameter in a Wedge
  • 4.3 Nonhomogeneous Energy Estimates for Solutions of Boundary Value Problems with Parameter in a Wedge
  • 4.3.1 Estimates on Solutions with Dirichlet BoundaryCondition
  • 4.3.2 Estimates on Solutions with Neumann Boundary Condition
  • 4.4 Strong Solutions
  • 4.4.1 The Dirichlet Problem with Homogeneous Energy Estimate in a Wedge
  • 4.4.2 The Dirichlet Problem with Nonhomogeneous Energy Estimate in a Wedge
  • 4.4.3 The Neumann Problem in a Wedge
  • 4.5 Weighted a priori Estimates for Solutions of Boundary Value Problems with Parameter in a Wedge
  • 4.5.1 Estimates of Solutions with DirichletBoundary Condition
  • 4.5.2 Estimate on Solutions with Neumann Boundary Condition in the Case dim K>2
  • 4.5.3 Estimates of Solutions with Neumann Boundary Condition for dim K=2

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