Regularity Techniques for Elliptic PDEs and the Fractional Laplacian PDF Download
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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: Pablo Raúl Stinga
Publisher: CRC Press
ISBN: 1040041574
Category : Mathematics
Languages : en
Pages : 923
Get Book Here
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: Pablo Raúl Stinga
Publisher: CRC Press
ISBN: 9781032679440
Category : Mathematics
Languages : en
Pages : 0
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Book Description
This book 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.
Author: Luis Angel Caffarelli
Publisher: Edizioni della Normale
ISBN: 9788876422492
Category : Mathematics
Languages : en
Pages : 0
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Book Description
The material presented here corresponds to Fermi lectures that I was invited to deliver at the Scuola Normale di Pisa in the spring of 1998. The obstacle problem consists in studying the properties of minimizers of the Dirichlet integral in a domain D of Rn, among all those configurations u with prescribed boundary values and costrained to remain in D above a prescribed obstacle F. In the Hilbert space H1(D) of all those functions with square integrable gradient, we consider the closed convex set K of functions u with fixed boundary value and which are greater than F in D. There is a unique point in K minimizing the Dirichlet integral. That is called the solution to the obstacle problem.
Author: Giovanni Molica Bisci
Publisher: Cambridge University Press
ISBN: 1107111943
Category : Mathematics
Languages : en
Pages : 401
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Book Description
A thorough graduate-level introduction to the variational analysis of nonlinear problems described by nonlocal operators.
Author: Jens M. Melenk
Publisher: Springer Nature
ISBN: 3031204328
Category : Mathematics
Languages : en
Pages : 571
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Book Description
The volume features high-quality papers based on the presentations at the ICOSAHOM 2020+1 on spectral and high order methods. The carefully reviewed articles cover state of the art topics in high order discretizations of partial differential equations. The volume presents a wide range of topics including the design and analysis of high order methods, the development of fast solvers on modern computer architecture, and the application of these methods in fluid and structural mechanics computations.
Author: Serena Dipierro
Publisher: Springer
ISBN: 303018921X
Category : Mathematics
Languages : en
Pages : 502
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Book Description
This volume collects contributions from the speakers at an INdAM Intensive period held at the University of Bari in 2017. The contributions cover several aspects of partial differential equations whose development in recent years has experienced major breakthroughs in terms of both theory and applications. The topics covered include nonlocal equations, elliptic equations and systems, fully nonlinear equations, nonlinear parabolic equations, overdetermined boundary value problems, maximum principles, geometric analysis, control theory, mean field games, and bio-mathematics. The authors are trailblazers in these topics and present their work in a way that is exhaustive and clearly accessible to PhD students and early career researcher. As such, the book offers an excellent introduction to a variety of fundamental topics of contemporary investigation and inspires novel and high-quality research.
Author: Serge Levendorskii
Publisher: Springer Science & Business Media
ISBN: 9401712158
Category : Mathematics
Languages : en
Pages : 442
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Book Description
This volume is the first to be devoted to the study of various properties of wide classes of degenerate elliptic operators of arbitrary order and pseudo-differential operators with multiple characteristics. Conditions for operators to be Fredholm in appropriate weighted Sobolev spaces are given, a priori estimates of solutions are derived, inequalities of the Grding type are proved, and the principal term of the spectral asymptotics for self-adjoint operators is computed. A generalization of the classical Weyl formula is proposed. Some results are new, even for operators of the second order. In addition, an analogue of the Boutet de Monvel calculus is developed and the index is computed. For postgraduate and research mathematicians, physicists and engineers whose work involves the solution of partial differential equations.
Author: C. Pozrikidis
Publisher: CRC Press
ISBN: 1315359936
Category : Mathematics
Languages : en
Pages : 396
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Book Description
The fractional Laplacian, also called the Riesz fractional derivative, describes an unusual diffusion process associated with random excursions. The Fractional Laplacian explores applications of the fractional Laplacian in science, engineering, and other areas where long-range interactions and conceptual or physical particle jumps resulting in an irregular diffusive or conductive flux are encountered. Presents the material at a level suitable for a broad audience of scientists and engineers with rudimentary background in ordinary differential equations and integral calculus Clarifies the concept of the fractional Laplacian for functions in one, two, three, or an arbitrary number of dimensions defined over the entire space, satisfying periodicity conditions, or restricted to a finite domain Covers physical and mathematical concepts as well as detailed mathematical derivations Develops a numerical framework for solving differential equations involving the fractional Laplacian and presents specific algorithms accompanied by numerical results in one, two, and three dimensions Discusses viscous flow and physical examples from scientific and engineering disciplines Written by a prolific author well known for his contributions in fluid mechanics, biomechanics, applied mathematics, scientific computing, and computer science, the book emphasizes fundamental ideas and practical numerical computation. It includes original material and novel numerical methods.
Author: Wenxiong Chen
Publisher: World Scientific
ISBN: 9813224010
Category : Mathematics
Languages : en
Pages : 342
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Book Description
This is a unique book that provides a comprehensive understanding of nonlinear equations involving the fractional Laplacian as well as other nonlocal operators. Beginning from the definition of fractional Laplacian, it gradually leads the readers to the frontier of current research in this area. The explanations and illustrations are elementary enough so that first year graduate students can follow easily, while it is advanced enough to include many new ideas, methods, and results that appeared recently in research literature, which researchers would find helpful. It focuses on introducing direct methods on the nonlocal problems without going through extensions, such as the direct methods of moving planes, direct method of moving spheres, direct blowing up and rescaling arguments, and so on. Different from most other books, it emphasizes on illuminating the ideas behind the formal concepts and proofs, so that readers can quickly grasp the essence.
Author: Susanne C. Brenner
Publisher: American Mathematical Soc.
ISBN: 1470451638
Category : Education
Languages : en
Pages : 378
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Book Description
The year 2018 marked the 75th anniversary of the founding of Mathematics of Computation, one of the four primary research journals published by the American Mathematical Society and the oldest research journal devoted to computational mathematics. To celebrate this milestone, the symposium “Celebrating 75 Years of Mathematics of Computation” was held from November 1–3, 2018, at the Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island. The sixteen papers in this volume, written by the symposium speakers and editors of the journal, include both survey articles and new contributions. On the discrete side, there are four papers covering topics in computational number theory and computational algebra. On the continuous side, there are twelve papers covering topics in machine learning, high dimensional approximations, nonlocal and fractional elliptic problems, gradient flows, hyperbolic conservation laws, Maxwell's equations, Stokes's equations, a posteriori error estimation, and iterative methods. Together they provide a snapshot of significant achievements in the past quarter century in computational mathematics and also in important current trends.
