Layer Potentials The Hodge Laplacian And Global Boundary Problems In Nonsmooth Riemannian Manifolds

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Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

Dorina Mitrea,Marius Mitrea,Michael Taylor
Layer Potentials the Hodge Laplacian and Global Boundary Problems in Nonsmooth Riemannian Manifolds Book PDF

Author : Dorina Mitrea,Marius Mitrea,Michael Taylor
Publisher : American Mathematical Soc.
Page : 137 pages
File Size : 55,5 Mb
Release : 2001
Category : Boundary value problems
ISBN : 9780821826591

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Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds by Dorina Mitrea,Marius Mitrea,Michael Taylor Pdf

The general aim of the present monograph is to study boundary-value problems for second-order elliptic operators in Lipschitz sub domains of Riemannian manifolds. In the first part (ss1-4), we develop a theory for Cauchy type operators on Lipschitz submanifolds of co dimension one (focused on boundedness properties and jump relations) and solve the $Lp$-Dirichlet problem, with $p$ close to $2$, for general second-order strongly elliptic systems. The solution is represented in the form of layer potentials and optimal non tangential maximal function estimates are established.This analysis is carried out under smoothness assumptions (for the coefficients of the operator, metric tensor and the underlying domain) which are in the nature of best possible. In the second part of the monograph, ss5-13, we further specialize this discussion to the case of Hodge Laplacian $\Delta: =-d\delta-\delta d$. This time, the goal is to identify all (pairs of) natural boundary conditions of Neumann type. Owing to the structural richness of the higher degree case we are considering, the theory developed here encompasses in a unitary fashion many basic PDE's of mathematical physics. Its scope extends to also cover Maxwell's equations, dealt with separately in s14. The main tools are those of PDE's and harmonic analysis, occasionally supplemented with some basic facts from algebraic topology and differential geometry.

The Hodge-Laplacian

Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor
The Hodge Laplacian Book PDF

Author : Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor
Publisher : Walter de Gruyter GmbH & Co KG
Page : 528 pages
File Size : 52,6 Mb
Release : 2016-10-10
Category : Mathematics
ISBN : 9783110484380

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The Hodge-Laplacian by Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor Pdf

The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains. Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals. Contents: Preface Introduction and Statement of Main Results Geometric Concepts and Tools Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism Additional Results and Applications Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis Bibliography Index

Harmonic Analysis and Boundary Value Problems

Luca Capogna,Loredana Lanzani
Harmonic Analysis and Boundary Value Problems Book PDF

Author : Luca Capogna,Loredana Lanzani
Publisher : American Mathematical Soc.
Page : 158 pages
File Size : 43,6 Mb
Release : 2001
Category : Mathematics
ISBN : 9780821827451

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Harmonic Analysis and Boundary Value Problems by Luca Capogna,Loredana Lanzani Pdf

This volume presents research and expository articles by the participants of the 25th Arkansas Spring Lecture Series on ``Recent Progress in the Study of Harmonic Measure from a Geometric and Analytic Point of View'' held at the University of Arkansas (Fayetteville). Papers in this volume provide clear and concise presentations of many problems that are at the forefront of harmonic analysis and partial differential equations. The following topics are featured: the solution of the Kato conjecture, the ``two bricks'' problem, new results on Cauchy integrals on non-smooth curves, the Neumann problem for sub-Laplacians, and a new general approach to both divergence and nondivergence second order parabolic equations based on growth theorems. The articles in this volume offer both students and researchers a comprehensive volume of current results in the field.

Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains

Mikhail S. Agranovich
Sobolev Spaces Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains Book PDF

Author : Mikhail S. Agranovich
Publisher : Springer
Page : 331 pages
File Size : 47,6 Mb
Release : 2015-05-06
Category : Mathematics
ISBN : 9783319146485

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Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains by Mikhail S. Agranovich Pdf

This book, which is based on several courses of lectures given by the author at the Independent University of Moscow, is devoted to Sobolev-type spaces and boundary value problems for linear elliptic partial differential equations. Its main focus is on problems in non-smooth (Lipschitz) domains for strongly elliptic systems. The author, who is a prominent expert in the theory of linear partial differential equations, spectral theory and pseudodifferential operators, has included his own very recent findings in the present book. The book is well suited as a modern graduate textbook, utilizing a thorough and clear format that strikes a good balance between the choice of material and the style of exposition. It can be used both as an introduction to recent advances in elliptic equations and boundary value problems and as a valuable survey and reference work. It also includes a good deal of new and extremely useful material not available in standard textbooks to date. Graduate and post-graduate students, as well as specialists working in the fields of partial differential equations, functional analysis, operator theory and mathematical physics will find this book particularly valuable.

Aspects of Boundary Problems in Analysis and Geometry

Juan Gil,Thomas Krainer,Ingo Witt
Aspects of Boundary Problems in Analysis and Geometry Book PDF

Author : Juan Gil,Thomas Krainer,Ingo Witt
Publisher : Birkhäuser
Page : 574 pages
File Size : 44,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034878500

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Aspects of Boundary Problems in Analysis and Geometry by Juan Gil,Thomas Krainer,Ingo Witt Pdf

Boundary problems constitute an essential field of common mathematical interest, they lie in the center of research activities both in analysis and geometry. This book encompasses material from both disciplines, and focuses on their interactions which are particularly apparent in this field. Moreover, the survey style of the contributions makes the topics accessible to a broad audience with a background in analysis or geometry, and enables the reader to get a quick overview.

Geometric Harmonic Analysis III

Dorina Mitrea,Irina Mitrea,Marius Mitrea
Geometric Harmonic Analysis III Book PDF

Author : Dorina Mitrea,Irina Mitrea,Marius Mitrea
Publisher : Springer Nature
Page : 980 pages
File Size : 54,7 Mb
Release : 2023-05-12
Category : Mathematics
ISBN : 9783031227356

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Geometric Harmonic Analysis III by Dorina Mitrea,Irina Mitrea,Marius Mitrea Pdf

This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations. Volume III is concerned with integral representation formulas for nullsolutions of elliptic PDEs, Calderón-Zygmund theory for singular integral operators, Fatou type theorems for systems of elliptic PDEs, and applications to acoustic and electromagnetic scattering. Overall, this amounts to a powerful and nuanced theory developed on uniformly rectifiable sets, which builds on the work of many predecessors.

Integral Methods in Science and Engineering

Christian Constanda,Andreas Kirsch
Integral Methods in Science and Engineering Book PDF

Author : Christian Constanda,Andreas Kirsch
Publisher : Birkhäuser
Page : 717 pages
File Size : 52,8 Mb
Release : 2015-10-13
Category : Mathematics
ISBN : 9783319167275

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Integral Methods in Science and Engineering by Christian Constanda,Andreas Kirsch Pdf

This contributed volume contains a collection of articles on state-of-the-art developments on the construction of theoretical integral techniques and their application to specific problems in science and engineering. Written by internationally recognized researchers, the chapters in this book are based on talks given at the Thirteenth International Conference on Integral Methods in Science and Engineering, held July 21–25, 2014, in Karlsruhe, Germany. A broad range of topics is addressed, from problems of existence and uniqueness for singular integral equations on domain boundaries to numerical integration via finite and boundary elements, conservation laws, hybrid methods, and other quadrature-related approaches. This collection will be of interest to researchers in applied mathematics, physics, and mechanical and electrical engineering, as well as graduate students in these disciplines and other professionals for whom integration is an essential tool.

Geometric Harmonic Analysis I

Dorina Mitrea,Irina Mitrea,Marius Mitrea
Geometric Harmonic Analysis I Book PDF

Author : Dorina Mitrea,Irina Mitrea,Marius Mitrea
Publisher : Springer Nature
Page : 940 pages
File Size : 48,5 Mb
Release : 2022-11-04
Category : Mathematics
ISBN : 9783031059506

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Geometric Harmonic Analysis I by Dorina Mitrea,Irina Mitrea,Marius Mitrea Pdf

This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations. Volume I establishes a sharp version of the Divergence Theorem (aka Fundamental Theorem of Calculus) which allows for an inclusive class of vector fields whose boundary trace is only assumed to exist in a nontangential pointwise sense.

Recent Trends in Operator Theory and Partial Differential Equations

Vladimir Maz'ya,David Natroshvili,Eugene Shargorodsky,Wolfgang L. Wendland
Recent Trends in Operator Theory and Partial Differential Equations Book PDF

Author : Vladimir Maz'ya,David Natroshvili,Eugene Shargorodsky,Wolfgang L. Wendland
Publisher : Birkhäuser
Page : 313 pages
File Size : 45,7 Mb
Release : 2017-02-23
Category : Mathematics
ISBN : 9783319470795

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Recent Trends in Operator Theory and Partial Differential Equations by Vladimir Maz'ya,David Natroshvili,Eugene Shargorodsky,Wolfgang L. Wendland Pdf

This volume is dedicated to the eminent Georgian mathematician Roland Duduchava on the occasion of his 70th birthday. It presents recent results on Toeplitz, Wiener-Hopf, and pseudodifferential operators, boundary value problems, operator theory, approximation theory, and reflects the broad spectrum of Roland Duduchava's research. The book is addressed to a wide audience of pure and applied mathematicians.

Tools for PDE

Michael E. Taylor
Tools for PDE Book PDF

Author : Michael E. Taylor
Publisher : American Mathematical Soc.
Page : 274 pages
File Size : 45,7 Mb
Release : 2000
Category : Differential equations, Partial
ISBN : 9780821843789

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Tools for PDE by Michael E. Taylor Pdf

Developing three related tools that are useful in the analysis of partial differential equations (PDEs) arising from the classical study of singular integral operators, this text considers pseudodifferential operators, paradifferential operators, and layer potentials.

Basic Global Relative Invariants for Homogeneous Linear Differential Equations

Roger Chalkley
Basic Global Relative Invariants for Homogeneous Linear Differential Equations Book PDF

Author : Roger Chalkley
Publisher : American Mathematical Soc.
Page : 223 pages
File Size : 45,5 Mb
Release : 2002
Category : Differential equations, Linear
ISBN : 9780821827819

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Basic Global Relative Invariants for Homogeneous Linear Differential Equations by Roger Chalkley Pdf

Given any fixed integer $m \ge 3$, the author presents simple formulas for $m - 2$ algebraically independent polynomials over $\mathbb{Q}$ having the remarkable property, with respect to transformations of homogeneous linear differential equations of order $m$, that each polynomial is both a semi-invariant of the first kind (with respect to changes of the dependent variable) and a semi-invariant of the second kind (with respect to changes of the independent variable). These relative invariants are suitable for global studies in several different contexts and do not require Laguerre-Forsyth reductions for their evaluation. In contrast, all of the general formulas for basic relative invariants that have been proposed by other researchers during the last 113 years are merely local ones that are either much too complicated or require a Laguerre-Forsyth reduction for each evaluation.

Multi-Interval Linear Ordinary Boundary Value Problems and Complex Symplectic Algebra

William Norrie Everitt,Lawrence Markus
Multi Interval Linear Ordinary Boundary Value Problems and Complex Symplectic Algebra Book PDF

Author : William Norrie Everitt,Lawrence Markus
Publisher : American Mathematical Soc.
Page : 79 pages
File Size : 48,6 Mb
Release : 2001
Category : Boundary value problems
ISBN : 9780821826690

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Multi-Interval Linear Ordinary Boundary Value Problems and Complex Symplectic Algebra by William Norrie Everitt,Lawrence Markus Pdf

A multi-interval quasi-differential system $\{I_{r},M_{r},w_{r}:r\in\Omega\}$ consists of a collection of real intervals, $\{I_{r}\}$, as indexed by a finite, or possibly infinite index set $\Omega$ (where $\mathrm{card} (\Omega)\geq\aleph_{0}$ is permissible), on which are assigned ordinary or quasi-differential expressions $M_{r}$ generating unbounded operators in the Hilbert function spaces $L_{r}^{2}\equiv L^{2}(I_{r};w_{r})$, where $w_{r}$ are given, non-negative weight functions. For each fixed $r\in\Omega$ assume that $M_{r}$ is Lagrange symmetric (formally self-adjoint) on $I_{r}$ and hence specifies minimal and maximal closed operators $T_{0,r}$ and $T_{1,r}$, respectively, in $L_{r}^{2}$. However the theory does not require that the corresponding deficiency indices $d_{r}^{-}$ and $d_{r}^{+}$ of $T_{0,r}$ are equal (e. g. the symplectic excess $Ex_{r}=d_{r}^{+}-d_{r}^{-}\neq 0$), in which case there will not exist any self-adjoint extensions of $T_{0,r}$ in $L_{r}^{2}$. In this paper a system Hilbert space $\mathbf{H}:=\sum_{r\,\in\,\Omega}\oplus L_{r}^{2}$ is defined (even for non-countable $\Omega$) with corresponding minimal and maximal system operators $\mathbf{T}_{0}$ and $\mathbf{T}_{1}$ in $\mathbf{H}$. Then the system deficiency indices $\mathbf{d}^{\pm} =\sum_{r\,\in\,\Omega}d_{r}^{\pm}$ are equal (system symplectic excess $Ex=0$), if and only if there exist self-adjoint extensions $\mathbf{T}$ of $\mathbf{T}_{0}$ in $\mathbf{H}$. The existence is shown of a natural bijective correspondence between the set of all such self-adjoint extensions $\mathbf{T}$ of $\mathbf{T}_{0}$, and the set of all complete Lagrangian subspaces $\mathsf{L}$ of the system boundary complex symplectic space $\mathsf{S}=\mathbf{D(T}_{1})/\mathbf{D(T}_{0})$. This result generalizes the earlier symplectic version of the celebrated GKN-Theorem for single interval systems to multi-interval systems. Examples of such complete Lagrangians, for both finite and infinite dimensional complex symplectic $\mathsf{S}$, illuminate new phenoma for the boundary value problems of multi-interval systems. These concepts have applications to many-particle systems of quantum mechanics, and to other physical problems.

Hardy Spaces and Potential Theory on $C^1$ Domains in Riemannian Manifolds

Martin Dindoš
Hardy Spaces and Potential Theory on C 1 Domains in Riemannian Manifolds Book PDF

Author : Martin Dindoš
Publisher : American Mathematical Soc.
Page : 78 pages
File Size : 51,8 Mb
Release : 2008
Category : Mathematics
ISBN : 9780821840436

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Hardy Spaces and Potential Theory on $C^1$ Domains in Riemannian Manifolds by Martin Dindoš Pdf

The author studies Hardy spaces on $C1$ and Lipschitz domains in Riemannian manifolds. Hardy spaces, originally introduced in 1920 in complex analysis setting, are invaluable tool in harmonic analysis. For this reason these spaces have been studied extensively by many authors.

Distributions, Partial Differential Equations, and Harmonic Analysis

Dorina Mitrea
Distributions Partial Differential Equations and Harmonic Analysis Book PDF

Author : Dorina Mitrea
Publisher : Springer
Page : 600 pages
File Size : 46,5 Mb
Release : 2018-12-29
Category : Mathematics
ISBN : 9783030032968

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Distributions, Partial Differential Equations, and Harmonic Analysis by Dorina Mitrea Pdf

The aim of this book is to offer, in a concise, rigorous, and largely self-contained manner, a rapid introduction to the theory of distributions and its applications to partial differential equations and harmonic analysis. The book is written in a format suitable for a graduate course spanning either over one-semester, when the focus is primarily on the foundational aspects, or over a two-semester period that allows for the proper amount of time to cover all intended applications as well. It presents a balanced treatment of the topics involved, and contains a large number of exercises (upwards of two hundred, more than half of which are accompanied by solutions), which have been carefully chosen to amplify the effect, and substantiate the power and scope, of the theory of distributions. Graduate students, professional mathematicians, and scientifically trained people with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. Throughout, a special effort has been made to develop the theory of distributions not as an abstract edifice but rather give the reader a chance to see the rationale behind various seemingly technical definitions, as well as the opportunity to apply the newly developed tools (in the natural build-up of the theory) to concrete problems in partial differential equations and harmonic analysis, at the earliest opportunity. The main additions to the current, second edition, pertain to fundamental solutions (through the inclusion of the Helmholtz operator, the perturbed Dirac operator, and their iterations) and the theory of Sobolev spaces (built systematically from the ground up, exploiting natural connections with the Fourier Analysis developed earlier in the monograph).

Boundary Integral Equations

George C. Hsiao,Wolfgang L. Wendland
Boundary Integral Equations Book PDF

Author : George C. Hsiao,Wolfgang L. Wendland
Publisher : Springer Nature
Page : 783 pages
File Size : 46,9 Mb
Release : 2021-03-26
Category : Mathematics
ISBN : 9783030711276

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Boundary Integral Equations by George C. Hsiao,Wolfgang L. Wendland Pdf

This is the second edition of the book which has two additional new chapters on Maxwell’s equations as well as a section on properties of solution spaces of Maxwell’s equations and their trace spaces. These two new chapters, which summarize the most up-to-date results in the literature for the Maxwell’s equations, are sufficient enough to serve as a self-contained introductory book on the modern mathematical theory of boundary integral equations in electromagnetics. The book now contains 12 chapters and is divided into two parts. The first six chapters present modern mathematical theory of boundary integral equations that arise in fundamental problems in continuum mechanics and electromagnetics based on the approach of variational formulations of the equations. The second six chapters present an introduction to basic classical theory of the pseudo-differential operators. The aforementioned corresponding boundary integral operators can now be recast as pseudo-differential operators. These serve as concrete examples that illustrate the basic ideas of how one may apply the theory of pseudo-differential operators and their calculus to obtain additional properties for the corresponding boundary integral operators. These two different approaches are complementary to each other. Both serve as the mathematical foundation of the boundary element methods, which have become extremely popular and efficient computational tools for boundary problems in applications. This book contains a wide spectrum of boundary integral equations arising in fundamental problems in continuum mechanics and electromagnetics. The book is a major scholarly contribution to the modern approaches of boundary integral equations, and should be accessible and useful to a large community of advanced graduate students and researchers in mathematics, physics, and engineering.