An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs

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Publisher : Springer Science & Business Media
ISBN 13 : 8876424431
Total Pages : 373 pages
Book Rating : 4.8/5 (764 download)

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Book Synopsis An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs by : Mariano Giaquinta

Download or read book An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs written by Mariano Giaquinta and published by Springer Science & Business Media. This book was released on 2013-07-30 with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume deals with the regularity theory for elliptic systems. We may find the origin of such a theory in two of the problems posed by David Hilbert in his celebrated lecture delivered during the International Congress of Mathematicians in 1900 in Paris: 19th problem: Are the solutions to regular problems in the Calculus of Variations always necessarily analytic? 20th problem: does any variational problem have a solution, provided that certain assumptions regarding the given boundary conditions are satisfied, and provided that the notion of a solution is suitably extended? During the last century these two problems have generated a great deal of work, usually referred to as regularity theory, which makes this topic quite relevant in many fields and still very active for research. However, the purpose of this volume, addressed mainly to students, is much more limited. We aim to illustrate only some of the basic ideas and techniques introduced in this context, confining ourselves to important but simple situations and refraining from completeness. In fact some relevant topics are omitted. Topics include: harmonic functions, direct methods, Hilbert space methods and Sobolev spaces, energy estimates, Schauder and L^p-theory both with and without potential theory, including the Calderon-Zygmund theorem, Harnack's and De Giorgi-Moser-Nash theorems in the scalar case and partial regularity theorems in the vector valued case; energy minimizing harmonic maps and minimal graphs in codimension 1 and greater than 1. In this second deeply revised edition we also included the regularity of 2-dimensional weakly harmonic maps, the partial regularity of stationary harmonic maps, and their connections with the case p=1 of the L^p theory, including the celebrated results of Wente and of Coifman-Lions-Meyer-Semmes.

Elliptic Regularity Theory

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Publisher : Springer
ISBN 13 : 3319274856
Total Pages : 214 pages
Book Rating : 4.3/5 (192 download)

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Book Synopsis Elliptic Regularity Theory by : Lisa Beck

Download or read book Elliptic Regularity Theory written by Lisa Beck and published by Springer. This book was released on 2016-04-08 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lecture notes provide a self-contained introduction to regularity theory for elliptic equations and systems in divergence form. After a short review of some classical results on everywhere regularity for scalar-valued weak solutions, the presentation focuses on vector-valued weak solutions to a system of several coupled equations. In the vectorial case, weak solutions may have discontinuities and so are expected, in general, to be regular only outside of a set of measure zero. Several methods are presented concerning the proof of such partial regularity results, and optimal regularity is discussed. Finally, a short overview is given on the current state of the art concerning the size of the singular set on which discontinuities may occur. The notes are intended for graduate and postgraduate students with a solid background in functional analysis and some familiarity with partial differential equations; they will also be of interest to researchers working on related topics.

Regularity Theory for Quasilinear Elliptic Systems and Monge - Ampere Equations in Two Dimensions

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Publisher : Springer
ISBN 13 : 3540466789
Total Pages : 137 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Regularity Theory for Quasilinear Elliptic Systems and Monge - Ampere Equations in Two Dimensions by : Friedmar Schulz

Download or read book Regularity Theory for Quasilinear Elliptic Systems and Monge - Ampere Equations in Two Dimensions written by Friedmar Schulz and published by Springer. This book was released on 2006-12-08 with total page 137 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lecture notes have been written as an introduction to the characteristic theory for two-dimensional Monge-Ampère equations, a theory largely developed by H. Lewy and E. Heinz which has never been presented in book form. An exposition of the Heinz-Lewy theory requires auxiliary material which can be found in various monographs, but which is presented here, in part because the focus is different, and also because these notes have an introductory character. Self-contained introductions to the regularity theory of elliptic systems, the theory of pseudoanalytic functions and the theory of conformal mappings are included. These notes grew out of a seminar given at the University of Kentucky in the fall of 1988 and are intended for graduate students and researchers interested in this area.

Regularity of Minimal Surfaces

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Publisher : Springer Science & Business Media
ISBN 13 : 3642117007
Total Pages : 634 pages
Book Rating : 4.6/5 (421 download)

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Book Synopsis Regularity of Minimal Surfaces by : Ulrich Dierkes

Download or read book Regularity of Minimal Surfaces written by Ulrich Dierkes and published by Springer Science & Business Media. This book was released on 2010-08-16 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau ́s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau ́s problem have no interior branch points.

Maximal Function Methods for Sobolev Spaces

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Publisher : American Mathematical Soc.
ISBN 13 : 1470465752
Total Pages : 354 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Maximal Function Methods for Sobolev Spaces by : Juha Kinnunen

Download or read book Maximal Function Methods for Sobolev Spaces written by Juha Kinnunen and published by American Mathematical Soc.. This book was released on 2021-08-02 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Contemporary Research in Elliptic PDEs and Related Topics

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Publisher : Springer
ISBN 13 : 303018921X
Total Pages : 502 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Contemporary Research in Elliptic PDEs and Related Topics by : Serena Dipierro

Download or read book Contemporary Research in Elliptic PDEs and Related Topics written by Serena Dipierro and published by Springer. This book was released on 2019-07-12 with total page 502 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Lectures on Elliptic Partial Differential Equations

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Publisher : Springer
ISBN 13 : 8876426515
Total Pages : 230 pages
Book Rating : 4.8/5 (764 download)

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Book Synopsis Lectures on Elliptic Partial Differential Equations by : Luigi Ambrosio

Download or read book Lectures on Elliptic Partial Differential Equations written by Luigi Ambrosio and published by Springer. This book was released on 2019-01-10 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book originates from the Elliptic PDE course given by the first author at the Scuola Normale Superiore in recent years. It covers the most classical aspects of the theory of Elliptic Partial Differential Equations and Calculus of Variations, including also more recent developments on partial regularity for systems and the theory of viscosity solutions.

A Course in the Calculus of Variations

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Publisher : Springer Nature
ISBN 13 : 3031450361
Total Pages : 354 pages
Book Rating : 4.0/5 (314 download)

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Book Synopsis A Course in the Calculus of Variations by : Filippo Santambrogio

Download or read book A Course in the Calculus of Variations written by Filippo Santambrogio and published by Springer Nature. This book was released on 2024-01-18 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to the broad topic of the calculus of variations. It addresses the most natural questions on variational problems and the mathematical complexities they present. Beginning with the scientific modeling that motivates the subject, the book then tackles mathematical questions such as the existence and uniqueness of solutions, their characterization in terms of partial differential equations, and their regularity. It includes both classical and recent results on one-dimensional variational problems, as well as the adaptation to the multi-dimensional case. Here, convexity plays an important role in establishing semi-continuity results and connections with techniques from optimization, and convex duality is even used to produce regularity results. This is then followed by the more classical Hölder regularity theory for elliptic PDEs and some geometric variational problems on sets, including the isoperimetric inequality and the Steiner tree problem. The book concludes with a chapter on the limits of sequences of variational problems, expressed in terms of Γ-convergence. While primarily designed for master's-level and advanced courses, this textbook, based on its author's instructional experience, also offers original insights that may be of interest to PhD students and researchers. A foundational understanding of measure theory and functional analysis is required, but all the essential concepts are reiterated throughout the book using special memo-boxes.

Global Analysis of Minimal Surfaces

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Publisher : Springer Science & Business Media
ISBN 13 : 3642117066
Total Pages : 547 pages
Book Rating : 4.6/5 (421 download)

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Book Synopsis Global Analysis of Minimal Surfaces by : Ulrich Dierkes

Download or read book Global Analysis of Minimal Surfaces written by Ulrich Dierkes and published by Springer Science & Business Media. This book was released on 2010-08-16 with total page 547 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many properties of minimal surfaces are of a global nature, and this is already true for the results treated in the first two volumes of the treatise. Part I of the present book can be viewed as an extension of these results. For instance, the first two chapters deal with existence, regularity and uniqueness theorems for minimal surfaces with partially free boundaries. Here one of the main features is the possibility of "edge-crawling" along free parts of the boundary. The third chapter deals with a priori estimates for minimal surfaces in higher dimensions and for minimizers of singular integrals related to the area functional. In particular, far reaching Bernstein theorems are derived. The second part of the book contains what one might justly call a "global theory of minimal surfaces" as envisioned by Smale. First, the Douglas problem is treated anew by using Teichmüller theory. Secondly, various index theorems for minimal theorems are derived, and their consequences for the space of solutions to Plateau ́s problem are discussed. Finally, a topological approach to minimal surfaces via Fredholm vector fields in the spirit of Smale is presented.

Homogenization Theory for Multiscale Problems

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Publisher : Springer Nature
ISBN 13 : 3031218337
Total Pages : 469 pages
Book Rating : 4.0/5 (312 download)

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Book Synopsis Homogenization Theory for Multiscale Problems by : Xavier Blanc

Download or read book Homogenization Theory for Multiscale Problems written by Xavier Blanc and published by Springer Nature. This book was released on 2023-04-29 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides a pedagogic and comprehensive introduction to homogenization theory with a special focus on problems set for non-periodic media. The presentation encompasses both deterministic and probabilistic settings. It also mixes the most abstract aspects with some more practical aspects regarding the numerical approaches necessary to simulate such multiscale problems. Based on lecture courses of the authors, the book is suitable for graduate students of mathematics and engineering.

Research in Mathematics of Materials Science

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Publisher : Springer Nature
ISBN 13 : 3031044967
Total Pages : 514 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Research in Mathematics of Materials Science by : Malena I. Español

Download or read book Research in Mathematics of Materials Science written by Malena I. Español and published by Springer Nature. This book was released on 2022-09-27 with total page 514 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume highlights contributions of women mathematicians in the study of complex materials and includes both original research papers and reviews. The featured topics and methods draw on the fields of Calculus of Variations, Partial Differential Equations, Functional Analysis, Differential Geometry and Topology, as well as Numerical Analysis and Mathematical Modelling. Areas of applications include foams, fluid-solid interactions, liquid crystals, shape-memory alloys, magnetic suspensions, failure in solids, plasticity, viscoelasticity, homogenization, crystallization, grain growth, and phase-field models.

Fractional Elliptic Problems with Critical Growth in the Whole of $\R^n$

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Publisher : Springer
ISBN 13 : 8876426019
Total Pages : 162 pages
Book Rating : 4.8/5 (764 download)

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Book Synopsis Fractional Elliptic Problems with Critical Growth in the Whole of $\R^n$ by : Serena Dipierro

Download or read book Fractional Elliptic Problems with Critical Growth in the Whole of $\R^n$ written by Serena Dipierro and published by Springer. This book was released on 2017-03-14 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lecture notes are devoted to the analysis of a nonlocal equation in the whole of Euclidean space. In studying this equation, all the necessary material is introduced in the most self-contained way possible, giving precise references to the literature when necessary. The results presented are original, but no particular prerequisite or knowledge of the previous literature is needed to read this text. The work is accessible to a wide audience and can also serve as introductory research material on the topic of nonlocal nonlinear equations.

Advances in Dynamic and Mean Field Games

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Publisher : Birkhäuser
ISBN 13 : 3319706195
Total Pages : 368 pages
Book Rating : 4.3/5 (197 download)

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Book Synopsis Advances in Dynamic and Mean Field Games by : Joseph Apaloo

Download or read book Advances in Dynamic and Mean Field Games written by Joseph Apaloo and published by Birkhäuser. This book was released on 2018-01-03 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: This contributed volume considers recent advances in dynamic games and their applications, based on presentations given at the 17th Symposium of the International Society of Dynamic Games, held July 12-15, 2016, in Urbino, Italy. Written by experts in their respective disciplines, these papers cover various aspects of dynamic game theory including mean-field games, stochastic and pursuit-evasion games, and computational methods for dynamic games. Topics covered include Pedestrian flow in crowded environments Models for climate change negotiations Nash Equilibria for dynamic games involving Volterra integral equations Differential games in healthcare markets Linear-quadratic Gaussian dynamic games Aircraft control in wind shear conditions Advances in Dynamic and Mean-Field Games presents state-of-the-art research in a wide spectrum of areas. As such, it serves as a testament to the continued vitality and growth of the field of dynamic games and their applications. It will be of interest to an interdisciplinary audience of researchers, practitioners, and graduate students.

Nonlinear Dispersive Waves and Fluids

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Publisher : American Mathematical Soc.
ISBN 13 : 1470441098
Total Pages : 275 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Nonlinear Dispersive Waves and Fluids by : Avy Soffer

Download or read book Nonlinear Dispersive Waves and Fluids written by Avy Soffer and published by American Mathematical Soc.. This book was released on 2019-03-12 with total page 275 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the AMS Special Session on Spectral Calculus and Quasilinear Partial Differential Equations and the AMS Special Session on PDE Analysis on Fluid Flows, which were held in January 2017 in Atlanta, Georgia. These two sessions shared the underlying theme of the analysis aspect of evolutionary PDEs and mathematical physics. The articles address the latest trends and perspectives in the area of nonlinear dispersive equations and fluid flows. The topics mainly focus on using state-of-the-art methods and techniques to investigate problems of depth and richness arising in quantum mechanics, general relativity, and fluid dynamics.

Multi-wave Medical Imaging: Mathematical Modelling And Imaging Reconstruction

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Publisher : World Scientific
ISBN 13 : 178634226X
Total Pages : 687 pages
Book Rating : 4.7/5 (863 download)

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Book Synopsis Multi-wave Medical Imaging: Mathematical Modelling And Imaging Reconstruction by : Hyeonbae Kang

Download or read book Multi-wave Medical Imaging: Mathematical Modelling And Imaging Reconstruction written by Hyeonbae Kang and published by World Scientific. This book was released on 2017-03-03 with total page 687 pages. Available in PDF, EPUB and Kindle. Book excerpt: Super-Resolution imaging refers to modern techniques of achieving resolution below conventional limits. This book gives a comprehensive overview of mathematical and computational techniques used to achieve this, providing a solid foundation on which to develop the knowledge and skills needed for practical application of techniques. Split into five parts, the first looks at the mathematical and probabilistic tools needed, before moving on to description of different types of imaging; single-wave, anomaly, multi-wave and spectroscopic and nanoparticle.As an important contribution to the understanding of super-resolution techniques in biomedical imaging, this book is a useful resource for scientists and engineers in the fields of biomedical imaging and super-resolution, and is self-contained reference for any newcomers to these fields.

Interpolation Theory

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Publisher : Springer
ISBN 13 : 8876426388
Total Pages : 208 pages
Book Rating : 4.8/5 (764 download)

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Book Synopsis Interpolation Theory by : Alessandra Lunardi

Download or read book Interpolation Theory written by Alessandra Lunardi and published by Springer. This book was released on 2018-05-05 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the third edition of the 1999 lecture notes of the courses on interpolation theory that the author delivered at the Scuola Normale in 1998 and 1999. In the mathematical literature there are many good books on the subject, but none of them is very elementary, and in many cases the basic principles are hidden below great generality. In this book the principles of interpolation theory are illustrated aiming at simplification rather than at generality. The abstract theory is reduced as far as possible, and many examples and applications are given, especially to operator theory and to regularity in partial differential equations. Moreover the treatment is self-contained, the only prerequisite being the knowledge of basic functional analysis.

Introduction to Stochastic Analysis and Malliavin Calculus

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Publisher : Springer
ISBN 13 : 8876424997
Total Pages : 286 pages
Book Rating : 4.8/5 (764 download)

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Book Synopsis Introduction to Stochastic Analysis and Malliavin Calculus by : Giuseppe Da Prato

Download or read book Introduction to Stochastic Analysis and Malliavin Calculus written by Giuseppe Da Prato and published by Springer. This book was released on 2014-07-01 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents an introductory course on differential stochastic equations and Malliavin calculus. The material of the book has grown out of a series of courses delivered at the Scuola Normale Superiore di Pisa (and also at the Trento and Funchal Universities) and has been refined over several years of teaching experience in the subject. The lectures are addressed to a reader who is familiar with basic notions of measure theory and functional analysis. The first part is devoted to the Gaussian measure in a separable Hilbert space, the Malliavin derivative, the construction of the Brownian motion and Itô's formula. The second part deals with differential stochastic equations and their connection with parabolic problems. The third part provides an introduction to the Malliavin calculus. Several applications are given, notably the Feynman-Kac, Girsanov and Clark-Ocone formulae, the Krylov-Bogoliubov and Von Neumann theorems. In this third edition several small improvements are added and a new section devoted to the differentiability of the Feynman-Kac semigroup is introduced. A considerable number of corrections and improvements have been made.