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Author: Augusto C. Ponce
Publisher: European Mathematical Society
ISBN: 9783037191408
Category : Differential equations, Elliptic
Languages : en
Pages : 468
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Book Description
Partial differential equations (PDEs) and geometric measure theory (GMT) are branches of analysis whose connections are usually not emphasized in introductory graduate courses. Yet one cannot dissociate the notions of mass or electric charge, naturally described in terms of measures, from the physical potential they generate. Having such a principle in mind, this book illustrates the beautiful interplay between tools from PDEs and GMT in a simple and elegant way by investigating properties such as existence and regularity of solutions of linear and nonlinear elliptic PDEs. Inspired by a variety of sources, from the pioneer balayage scheme of Poincare to more recent results related to the Thomas-Fermi and Chern-Simons models, the problems covered in this book follow an original presentation, intended to emphasize the main ideas in the proofs. Classical techniques such as regularity theory, maximum principles and the method of sub- and supersolutions are adapted to the setting where merely integrability or density assumptions on the data are available. The distinguished role played by capacities and precise representatives is also explained. Other special features are: the remarkable equivalence between Sobolev capacities and Hausdorff contents in terms of trace inequalities; the strong approximation of measures in terms of capacities or densities, normally absent from GMT books; and the rescue of the strong maximum principle for the Schrodinger operator involving singular potentials. This book invites the reader on a trip through modern techniques in the frontier of elliptic PDEs and GMT and is addressed to graduate students and researchers with a deep interest in analysis. Most of the chapters can be read independently, and only a basic knowledge of measure theory, functional analysis, and Sobolev spaces is required.
Author: Augusto C. Ponce
Publisher: European Mathematical Society
ISBN: 9783037191408
Category : Differential equations, Elliptic
Languages : en
Pages : 468
Get Book Here
Book Description
Partial differential equations (PDEs) and geometric measure theory (GMT) are branches of analysis whose connections are usually not emphasized in introductory graduate courses. Yet one cannot dissociate the notions of mass or electric charge, naturally described in terms of measures, from the physical potential they generate. Having such a principle in mind, this book illustrates the beautiful interplay between tools from PDEs and GMT in a simple and elegant way by investigating properties such as existence and regularity of solutions of linear and nonlinear elliptic PDEs. Inspired by a variety of sources, from the pioneer balayage scheme of Poincare to more recent results related to the Thomas-Fermi and Chern-Simons models, the problems covered in this book follow an original presentation, intended to emphasize the main ideas in the proofs. Classical techniques such as regularity theory, maximum principles and the method of sub- and supersolutions are adapted to the setting where merely integrability or density assumptions on the data are available. The distinguished role played by capacities and precise representatives is also explained. Other special features are: the remarkable equivalence between Sobolev capacities and Hausdorff contents in terms of trace inequalities; the strong approximation of measures in terms of capacities or densities, normally absent from GMT books; and the rescue of the strong maximum principle for the Schrodinger operator involving singular potentials. This book invites the reader on a trip through modern techniques in the frontier of elliptic PDEs and GMT and is addressed to graduate students and researchers with a deep interest in analysis. Most of the chapters can be read independently, and only a basic knowledge of measure theory, functional analysis, and Sobolev spaces is required.
Author: Serena Dipierro
Publisher: World Scientific
ISBN: 9811290814
Category : Mathematics
Languages : en
Pages : 670
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Book Description
This is a textbook that covers several selected topics in the theory of elliptic partial differential equations which can be used in an advanced undergraduate or graduate course.The book considers many important issues such as existence, regularity, qualitative properties, and all the classical topics useful in the wide world of partial differential equations. It also includes applications with interesting examples.The structure of the book is flexible enough to allow different chapters to be taught independently.The book is friendly, welcoming, and written for a newcomer to the subject.It is essentially self-contained, making it easy to read, and all the concepts are fully explained from scratch, combining intuition and rigor, and therefore it can also be read independently by students, with limited or no supervision.
Author: Ireneo Peral Alonso
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110605600
Category : Mathematics
Languages : en
Pages : 406
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Book Description
The scientific literature on the Hardy-Leray inequality, also known as the uncertainty principle, is very extensive and scattered. The Hardy-Leray potential shows an extreme spectral behavior and a peculiar influence on diffusion problems, both stationary and evolutionary. In this book, a big part of the scattered knowledge about these different behaviors is collected in a unified and comprehensive presentation.
Author: Suzanne Lenhart
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110792788
Category : Mathematics
Languages : en
Pages : 365
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Book Description
This volume is dedicated to the legacy of David R. Adams (1941-2021) and discusses calculus of variations, functional - harmonic - potential analysis, partial differential equations, and their applications in modeling, mathematical physics, and differential - integral geometry.
Author: Pablo Raúl Stinga
Publisher: CRC Press
ISBN: 1040041574
Category : Mathematics
Languages : en
Pages : 923
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Book Description
Regularity Techniques for Elliptic PDEs and the Fractional Laplacian presents important analytic and geometric techniques to prove regularity estimates for solutions to second order elliptic equations, both in divergence and nondivergence form, and to nonlocal equations driven by the fractional Laplacian. The emphasis is placed on ideas and the development of intuition, while at the same time being completely rigorous. The reader should keep in mind that this text is about how analysis can be applied to regularity estimates. Many methods are nonlinear in nature, but the focus is on linear equations without lower order terms, thus avoiding bulky computations. The philosophy underpinning the book is that ideas must be flushed out in the cleanest and simplest ways, showing all the details and always maintaining rigor. Features Self-contained treatment of the topic Bridges the gap between upper undergraduate textbooks and advanced monographs to offer a useful, accessible reference for students and researchers. Replete with useful references.
Author: Xavier Fernández-Real
Publisher: Springer Nature
ISBN: 3031542428
Category : Differential equations, Elliptic
Languages : en
Pages : 409
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Book Description
Zusammenfassung: This monograph offers a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. This class of equations finds relevance in fields such as analysis, probability theory, mathematical physics, and in several contexts in the applied sciences. The work gives a detailed presentation of all the necessary techniques, with a primary focus on the main ideas rather than on proving all the results in their greatest generality. The basic building blocks are presented first, with the study of the square root of the Laplacian, and weak solutions to linear equations. Subsequently, the theory of viscosity solutions to nonlinear equations is developed, and proofs are provided for the main known results in this context. The analysis finishes with the investigation of obstacle problems for integro-differential operators and establishes the regularity of solutions and free boundaries. A distinctive feature of this work lies in its presentation of nearly all covered material in a monographic format for the first time, and several proofs streamline, and often simplify, those in the original papers. Furthermore, various open problems are listed throughout the chapters
Author:
Publisher: Springer Nature
ISBN: 3031709098
Category :
Languages : en
Pages : 439
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Book Description
Author: Shouhei Honda
Publisher: American Mathematical Soc.
ISBN: 1470428547
Category : Mathematics
Languages : en
Pages : 104
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Book Description
In this paper the author studies elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular the author establishes continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrödinger operators, generalized Yamabe constants and eigenvalues of the Hodge Laplacian, with respect to the Gromov-Hausdorff topology. The author applies these to the study of second-order differential calculus on such limit spaces.
Author: Juha Kinnunen
Publisher: American Mathematical Soc.
ISBN: 1470465752
Category : Education
Languages : en
Pages : 354
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Book Description
This book discusses advances in maximal function methods related to Poincaré and Sobolev inequalities, pointwise estimates and approximation for Sobolev functions, Hardy's inequalities, and partial differential equations. Capacities are needed for fine properties of Sobolev functions and characterization of Sobolev spaces with zero boundary values. The authors consider several uniform quantitative conditions that are self-improving, such as Hardy's inequalities, capacity density conditions, and reverse Hölder inequalities. They also study Muckenhoupt weight properties of distance functions and combine these with weighted norm inequalities; notions of dimension are then used to characterize density conditions and to give sufficient and necessary conditions for Hardy's inequalities. At the end of the book, the theory of weak solutions to the p p-Laplace equation and the use of maximal function techniques is this context are discussed. The book is directed to researchers and graduate students interested in applications of geometric and harmonic analysis in Sobolev spaces and partial differential equations.
Author: Craig C. Douglas
Publisher: SIAM
ISBN: 0898715415
Category : Technology & Engineering
Languages : en
Pages : 146
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Book Description
A Tutorial on Elliptic PDE Solvers and Their Parallelization is a valuable aid for learning about the possible errors and bottlenecks in parallel computing. One of the highlights of the tutorial is that the course material can run on a laptop, not just on a parallel computer or cluster of PCs, thus allowing readers to experience their first successes in parallel computing in a relatively short amount of time. This tutorial is intended for advanced undergraduate and graduate students in computational sciences and engineering; however, it may also be helpful to professionals who use PDE-based parallel computer simulations in the field.
