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Asymptotics of Elliptic and Parabolic PDEs

Author : David Holcman

Publisher : Springer

Page : 456 pages

File Size : 36,59 MB

Release : 2018-05-25

Category : Mathematics

ISBN : 3319768956

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Book Description

This is a monograph on the emerging branch of mathematical biophysics combining asymptotic analysis with numerical and stochastic methods to analyze partial differential equations arising in biological and physical sciences. In more detail, the book presents the analytic methods and tools for approximating solutions of mixed boundary value problems, with particular emphasis on the narrow escape problem. Informed throughout by real-world applications, the book includes topics such as the Fokker-Planck equation, boundary layer analysis, WKB approximation, applications of spectral theory, as well as recent results in narrow escape theory. Numerical and stochastic aspects, including mean first passage time and extreme statistics, are discussed in detail and relevant applications are presented in parallel with the theory. Including background on the classical asymptotic theory of differential equations, this book is written for scientists of various backgrounds interested in deriving solutions to real-world problems from first principles.





Recent Advances on Elliptic and Parabolic Issues

Author : Michel Chipot

Publisher : World Scientific

Page : 302 pages

File Size : 42,80 MB

Release : 2006

Category : Mathematics

ISBN : 9812774173

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Book Description

This volume is a collection of articles discussing the most recent advances on various topics in partial differential equations. Many important issues regarding evolution problems, their asymptotic behavior and their qualitative properties are addressed. The quality and completeness of the articles will make this book a source of inspiration and references in the future. Contents: Steady Free Convection in a Bounded and Saturated Porous Medium (S Akesbi et al.); Quasilinear Parabolic Functional Evolution Equations (H Amann); A Linear Parabolic Problem with Non-Dissipative Dynamical Boundary Conditions (C Bandle & W Reichel); Remarks on Some Class of Nonlocal Elliptic Problems (M Chipot); On Some Definitions and Properties of Generalized Convex Sets Arising in the Calculus of Variations (B Dacorogna et al.); Note on the Asymptotic Behavior of Solutions to an Anisotropic Crystalline Curvature Flow (C Hirota et al.); A Reaction-Diffusion Approximation to a Cross-Diffusion System (M Iida et al.); Bifurcation Diagrams to an Elliptic Equation Involving the Critical Sobolev Exponent with the Robin Condition (Y Kabeya); Ginzburg-Landau Functional in a Thin Loop and Local Minimizers (S Kosugi & Y Morita); Singular Limit for Some Reaction Diffusion System (K Nakashima); Rayleigh-B(r)nard Convection in a Rectangular Domain (T Ogawa & T Okuda); Some Convergence Results for Elliptic Problems with Periodic Data (Y Xie); On Global Unbounded Solutions for a Semilinear Parabolic Equation (E Yanagida). Readership: Graduate students and researchers in partial differential equations and nonlinear science.



Elliptic And Parabolic Equations

Author : Zhuoqun Wu

Publisher : World Scientific Publishing Company

Page : 425 pages

File Size : 18,10 MB

Release : 2006-10-17

Category : Mathematics

ISBN : 9813101709

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Book Description

This book provides an introduction to elliptic and parabolic equations. While there are numerous monographs focusing separately on each kind of equations, there are very few books treating these two kinds of equations in combination. This book presents the related basic theories and methods to enable readers to appreciate the commonalities between these two kinds of equations as well as contrast the similarities and differences between them.



Recent Advances on Elliptic and Parabolic Issues

Author : Michel Chipot

Publisher : World Scientific Publishing Company Incorporated

Page : 292 pages

File Size : 11,69 MB

Release : 2006

Category : Mathematics

ISBN : 9789812566751

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Book Description

This volume is a collection of articles discussing the most recent advances on various topics in partial differential equations. Many important issues regarding evolution problems, their asymptotic behavior and their qualitative properties are addressed. The quality and completeness of the articles will make this book a source of inspiration and references in the future. Contents: Steady Free Convection in a Bounded and Saturated Porous Medium (S Akesbi et al.); Quasilinear Parabolic Functional Evolution Equations (H Amann); A Linear Parabolic Problem with Non-Dissipative Dynamical Boundary Conditions (C Bandle & W Reichel); Remarks on Some Class of Nonlocal Elliptic Problems (M Chipot); On Some Definitions and Properties of Generalized Convex Sets Arising in the Calculus of Variations (B Dacorogna et al.); Note on the Asymptotic Behavior of Solutions to an Anisotropic Crystalline Curvature Flow (C Hirota et al.); A Reaction-Diffusion Approximation to a Cross-Diffusion System (M Iida et al.); Bifurcation Diagrams to an Elliptic Equation Involving the Critical Sobolev Exponent with the Robin Condition (Y Kabeya); Ginzburg-Landau Functional in a Thin Loop and Local Minimizers (S Kosugi & Y Morita); Singular Limit for Some Reaction Diffusion System (K Nakashima); Rayleigh-Benard Convection in a Rectangular Domain (T Ogawa & T Okuda); Some Convergence Results for Elliptic Problems with Periodic Data (Y Xie); On Global Unbounded Solutions for a Semilinear Parabolic Equation (E Yanagida). Key Features An accessible presentation of the latest, cutting-edge topics in partial differential equations Written by leading scholars in related fields Readership: Graduate students and researchers in partial differential equations and nonlinear science.



Elliptic and Parabolic Equations with Discontinuous Coefficients

Author : Antonino Maugeri

Publisher : Wiley-VCH

Page : 266 pages

File Size : 46,93 MB

Release : 2000-12-13

Category : Mathematics

ISBN :

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Book Description

This book unifies the different approaches in studying elliptic and parabolic partial differential equations with discontinuous coefficients. To the enlarging market of researchers in applied sciences, mathematics and physics, it gives concrete answers to questions suggested by non-linear models. Providing an up-to date survey on the results concerning elliptic and parabolic operators on a high level, the authors serve the reader in doing further research. Being themselves active researchers in the field, the authors describe both on the level of good examples and precise analysis, the crucial role played by such requirements on the coefficients as the Cordes condition, Campanato's nearness condition, and vanishing mean oscillation condition. They present the newest results on the basic boundary value problems for operators with VMO coefficients and non-linear operators with discontinuous coefficients and state a lot of open problems in the field.



Parabolicity, Volterra Calculus, and Conical Singularities

Author : Sergio Albeverio

Publisher : Springer Science & Business Media

Page : 380 pages

File Size : 49,49 MB

Release : 2002

Category : Mathematics

ISBN : 9783764369064

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Book Description

Volterra Families of Pseudodifferential Operators.- 1. Basic notation and general conventions.- 1.1. Sets of real and complex numbers.- 1.2. Multi-index notation.- 1.3. Functional analysis and basic function spaces.- 1.4. Tempered distributions and the Fourier transform.- 2. General parameter-dependent symbols.- 2.1. Asymptotic expansion.- 2.2. Homogeneity and classical symbols.- 3. Parameter-dependent Volterra symbols.- 3.1. Kernel cut-off and asymptotic expansion.- 3.2. The translation operator in Volterra symbols.- 4. The calculus of pseudodifferential operators.- 4.1. Elements of the calculus.- 4.2. The formal adjoint operator.- 4.3. Sobolev spaces and continuity.- 4.4. Coordinate invariance.- 5. Ellipticity and parabolicity.- 5.1. Ellipticity in the general calculus.- 5.2. Parabolicity in the Volterra calculus.- References.- The Calculus of Volterra Mellin Pseudodifferential Operators with Operator-valued Symbols.- 1. Preliminaries on function spaces and the Mellin transform.- 1.1. A Paley-Wiener type theorem.- 1.2. The Mellin transform in distributions.- 2. The calculus of Volterra symbols.- 2.1. General anisotropic and Volterra symbols.- 2.1.1. Hilbert spaces with group-actions.- 2.1.2. Definition of the symbol spaces.- 2.1.3. Asymptotic expansion.- 2.1.4. The translation operator in Volterra symbols.- 2.2. Holomorphic Volterra symbols.- 3. The calculus of Volterra Mellin operators.- 3.1. General Volterra Mellin operators.- 3.2. Continuity in Mellin Sobolev spaces.- 3.3. Volterra Mellin operators with analytic symbols.- 4. Kernel cut-off and Mellin quantization.- 4.1. The Mellin kernel cut-off operator.- 4.2. Degenerate symbols and Mellin quantization.- 5. Parabolicity and Volterra parametrices.- 5.1. Ellipticity and parabolicity on symbolic level.- 5.2. The parametrix construction.- References.- On the Inverse of Parabolic Systems of Partial Differential Equations of General Form in an Infinite Space-Time Cylinder.- 1. Preliminary material.- 1.1. Basic notation and general conventions.- Functional analysis and basic function spaces.- Preliminaries on function spaces and the Mellin transform.- Global analysis.- 1.2. Finitely meromorphic Fredholm families in ?-algebras.- 1.3. Volterra integral operators.- Some notes on abstract kernels.- 2. Abstract Volterra pseudodifferential calculus.- 2.1. Anisotropic parameter-dependent symbols.- Asymptotic expansion.- Classical symbols.- 2.2. Anisotropic parameter-dependent operators.- Elements of the calculus.- Ellipticity and parametrices.- Sobolev spaces and continuity.- Coordinate invariance.- 2.3. Parameter-dependent Volterra symbols.- Kernel cut-off and asymptotic expansion of Volterra symbols.- The translation operator in Volterra symbols.- 2.4. Parameter-dependent Volterra operators.- Elements of the calculus.- Continuity and coordinate invariance.- Parabolicity for Volterra pseudodifferential operators.- 2.5. Volterra Mellin calculus.- Continuity in Mellin Sobolev spaces.- 2.6. Analytic Volterra Mellin calculus.- Elements of the calculus.- The Mellin kernel cut-off operator and asymptotic expansion.- Degenerate symbols and Mellin quantization.- 2.7. Volterra Fourier operators with global weight conditions.- 3. Parameter-dependent Volterra calculus on a closed manifold.- 3.1. Anisotropic parameter-dependent operators.- Ellipticity and parametrices.- 3.2. Parameter-dependent Volterra operators.- Kernel cut-off behaviour and asymptotic expansion.- The translation operator in Volterra pseudodifferential operators.- Parabolicity for Volterra operators on manifolds.- 4. Weighted Sobolev spaces.- 4.1. Anisotropic Sobolev spaces on the infinite cylinder.- 4.2. Anisotropic Mellin Sobolev spaces.- Mellin Sobolev spaces with asymptotics.- 4.3. Cone Sobolev spaces.- 5. Calculi built upon parameter-dependent operators.- 5.1. Anisotropic meromorphic Mellin symbols.- 5.2. Meromorphic Volterra Mellin symbols.- Mellin quantization.- 5.3. Elements of the Mellin calculus.- Ellipticity and ...



Elliptic and Parabolic Equations

Author : Joachim Escher

Publisher : Springer

Page : 295 pages

File Size : 15,83 MB

Release : 2015-06-04

Category : Mathematics

ISBN : 3319125478

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Book Description

The international workshop on which this proceedings volume is based on brought together leading researchers in the field of elliptic and parabolic equations. Particular emphasis was put on the interaction between well-established scientists and emerging young mathematicians, as well as on exploring new connections between pure and applied mathematics. The volume contains material derived after the workshop taking up the impetus to continue collaboration and to incorporate additional new results and insights.



Recent Advances on Elliptic and Parabolic Issues

Author : Michel Chipot

Publisher : World Scientific

Page : 302 pages

File Size : 49,20 MB

Release : 2006

Category : Mathematics

ISBN : 9812566759

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Book Description

This volume is a collection of articles discussing the most recent advances on various topics in partial differential equations. Many important issues regarding evolution problems, their asymptotic behavior and their qualitative properties are addressed. The quality and completeness of the articles will make this book a source of inspiration and references in the future. Contents: Steady Free Convection in a Bounded and Saturated Porous Medium (S Akesbi et al.); Quasilinear Parabolic Functional Evolution Equations (H Amann); A Linear Parabolic Problem with Non-Dissipative Dynamical Boundary Conditions (C Bandle & W Reichel); Remarks on Some Class of Nonlocal Elliptic Problems (M Chipot); On Some Definitions and Properties of Generalized Convex Sets Arising in the Calculus of Variations (B Dacorogna et al.); Note on the Asymptotic Behavior of Solutions to an Anisotropic Crystalline Curvature Flow (C Hirota et al.); A Reaction-Diffusion Approximation to a Cross-Diffusion System (M Iida et al.); Bifurcation Diagrams to an Elliptic Equation Involving the Critical Sobolev Exponent with the Robin Condition (Y Kabeya); Ginzburg-Landau Functional in a Thin Loop and Local Minimizers (S Kosugi & Y Morita); Singular Limit for Some Reaction Diffusion System (K Nakashima); Rayleigh-Benard Convection in a Rectangular Domain (T Ogawa & T Okuda); Some Convergence Results for Elliptic Problems with Periodic Data (Y Xie); On Global Unbounded Solutions for a Semilinear Parabolic Equation (E Yanagida). Key Features An accessible presentation of the latest, cutting-edge topics in partial differential equations Written by leading scholars in related fields Readership: Graduate students and researchers in partial differential equations and nonlinear science.



Lectures on Elliptic and Parabolic Equations in Holder Spaces

Author : Nikolaĭ Vladimirovich Krylov

Publisher : American Mathematical Soc.

Page : 178 pages

File Size : 29,91 MB

Release : 1996

Category : Mathematics

ISBN : 082180569X

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Book Description

These lectures concentrate on fundamentals of the modern theory of linear elliptic and parabolic equations in H older spaces. Krylov shows that this theory - including some issues of the theory of nonlinear equations - is based on some general and extremely powerful ideas and some simple computations. The main object of study is the first boundary-value problems for elliptic and parabolic equations, with some guidelines concerning other boundary-value problems such as the Neumann or oblique derivative problems or problems involving higher-order elliptic operators acting on the boundary. Numerical approximations are also discussed. This book, containing 200 exercises, aims to provide a good understanding of what kind of results are available and what kinds of techniques are used to obtain them.