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

Author: Dorina Mitrea

Publisher: American Mathematical Soc.

Published: 2001

Total Pages: 137

ISBN-13: 082182659X

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.

Geometric Harmonic Analysis III

Author: Dorina Mitrea

Publisher: Springer Nature

Published: 2023-05-12

Total Pages: 980

ISBN-13: 3031227352

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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.

The Hodge-Laplacian

Author: Dorina Mitrea

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2016-10-10

Total Pages: 528

ISBN-13: 3110484382

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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

Geometric Harmonic Analysis I

Author: Dorina Mitrea

Publisher: Springer Nature

Published: 2022-11-04

Total Pages: 940

ISBN-13: 3031059506

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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.

Aspects of Boundary Problems in Analysis and Geometry

Author: Juan Gil

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 574

ISBN-13: 3034878508

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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.

Topics in Mathematical Analysis and Applications

Author: Themistocles M. Rassias

Publisher: Springer

Published: 2014-10-13

Total Pages: 811

ISBN-13: 3319065548

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This volume presents significant advances in a number of theories and problems of Mathematical Analysis and its applications in disciplines such as Analytic Inequalities, Operator Theory, Functional Analysis, Approximation Theory, Functional Equations, Differential Equations, Wavelets, Discrete Mathematics and Mechanics. The contributions focus on recent developments and are written by eminent scientists from the international mathematical community. Special emphasis is given to new results that have been obtained in the above mentioned disciplines in which Nonlinear Analysis plays a central role. Some review papers published in this volume will be particularly useful for a broader readership in Mathematical Analysis, as well as for graduate students. An attempt is given to present all subjects in this volume in a unified and self-contained manner, to be particularly useful to the mathematical community.

Harmonic Analysis and Boundary Value Problems

Author: Luca Capogna

Publisher: American Mathematical Soc.

Published: 2001

Total Pages: 170

ISBN-13: 0821827456

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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.

Semigroups of Operators: Theory and Applications

Author: A.V. Balakrishnan

Publisher: Springer Science & Business Media

Published: 2000-08-01

Total Pages: 386

ISBN-13: 9783764363109

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These Proceedings comprise the bulk of the papers presented at the Inter national Conference on Semigroups of Opemtors: Theory and Contro~ held 14-18 December 1998, Newport Beach, California, U.S.A. The intent of the Conference was to highlight recent advances in the the ory of Semigroups of Operators which provides the abstract framework for the time-domain solutions of time-invariant boundary-value/initial-value problems of partial differential equations. There is of course a firewall between the ab stract theory and the applications and one of the Conference aims was to bring together both in the hope that it may be of value to both communities. In these days when all scientific activity is judged by its value on "dot com" it is not surprising that mathematical analysis that holds no promise of an immediate commercial product-line, or even a software tool-box, is not high in research priority. We are particularly pleased therefore that the National Science Foundation provided generous financial support without which this Conference would have been impossible to organize. Our special thanks to Dr. Kishan Baheti, Program Manager.

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

Author: Mikhail S. Agranovich

Publisher: Springer

Published: 2015-05-06

Total Pages: 343

ISBN-13: 3319146483

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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.

Tools for PDE

Author: Michael E. Taylor

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 274

ISBN-13: 0821843788

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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.