An Introduction To The Regularity Theory For Elliptic Systems Harmonic Maps And Minimal Graphs

Author : Mariano Giaquinta,Luca Martinazzi
Publisher : Springer Science & Business Media
Page : 370 pages
File Size : 46,7 Mb
Release : 2013-07-30
Category : Mathematics
ISBN : 9788876424434

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An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs by Mariano Giaquinta,Luca Martinazzi Pdf

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.

Author : James Eells,Luc Lemaire
Publisher : World Scientific
Page : 38 pages
File Size : 40,5 Mb
Release : 1995
Category : Mathematics
ISBN : 9810214669

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Two Reports on Harmonic Maps by James Eells,Luc Lemaire Pdf

Harmonic maps between Riemannian manifolds are solutions of systems of nonlinear partial differential equations which appear in different contexts of differential geometry. They include holomorphic maps, minimal surfaces, å-models in physics. Recently, they have become powerful tools in the study of global properties of Riemannian and K„hlerian manifolds.A standard reference for this subject is a pair of Reports, published in 1978 and 1988 by James Eells and Luc Lemaire.This book presents these two reports in a single volume with a brief supplement reporting on some recent developments in the theory. It is both an introduction to the subject and a unique source of references, providing an organized exposition of results spread throughout more than 800 papers.

Author : Lisa Beck
Publisher : Springer
Page : 201 pages
File Size : 51,5 Mb
Release : 2016-04-08
Category : Mathematics
ISBN : 9783319274850

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Elliptic Regularity Theory by Lisa Beck Pdf

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.

Author : Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba
Publisher : Springer Science & Business Media
Page : 634 pages
File Size : 47,5 Mb
Release : 2010-08-16
Category : Mathematics
ISBN : 9783642117008

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Regularity of Minimal Surfaces by Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba Pdf

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.

Author : Mariano Giaquinta
Publisher : Birkhauser
Page : 152 pages
File Size : 42,7 Mb
Release : 1993
Category : Mathematics
ISBN : UOM:39015028915448

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Introduction to Regularity Theory for Nonlinear Elliptic Systems by Mariano Giaquinta Pdf

Author : Juha Kinnunen,Juha Lehrbäck,Antti Vähäkangas
Publisher : American Mathematical Soc.
Page : 354 pages
File Size : 46,7 Mb
Release : 2021-08-02
Category : Education
ISBN : 9781470465759

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Maximal Function Methods for Sobolev Spaces by Juha Kinnunen,Juha Lehrbäck,Antti Vähäkangas Pdf

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 : Luigi Ambrosio,Alessandro Carlotto,Annalisa Massaccesi
Publisher : Springer
Page : 230 pages
File Size : 48,8 Mb
Release : 2019-01-10
Category : Mathematics
ISBN : 9788876426513

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Lectures on Elliptic Partial Differential Equations by Luigi Ambrosio,Alessandro Carlotto,Annalisa Massaccesi Pdf

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.

Author : Zhongwei Shen
Publisher : Springer
Page : 291 pages
File Size : 46,5 Mb
Release : 2018-09-04
Category : Mathematics
ISBN : 9783319912141

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Periodic Homogenization of Elliptic Systems by Zhongwei Shen Pdf

This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization.

Author : Xavier Blanc,Claude Le Bris
Publisher : Springer Nature
Page : 469 pages
File Size : 47,9 Mb
Release : 2023-04-29
Category : Mathematics
ISBN : 9783031218330

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Homogenization Theory for Multiscale Problems by Xavier Blanc,Claude Le Bris Pdf

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.

Author : Malena I. Español,Marta Lewicka,Lucia Scardia,Anja Schlömerkemper
Publisher : Springer Nature
Page : 514 pages
File Size : 50,9 Mb
Release : 2022-09-27
Category : Mathematics
ISBN : 9783031044960

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Research in Mathematics of Materials Science by Malena I. Español,Marta Lewicka,Lucia Scardia,Anja Schlömerkemper Pdf

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.

Author : Serena Dipierro,María Medina,Enrico Valdinoci
Publisher : Springer
Page : 155 pages
File Size : 45,9 Mb
Release : 2017-03-14
Category : Mathematics
ISBN : 9788876426018

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Fractional Elliptic Problems with Critical Growth in the Whole of $\R^n$ by Serena Dipierro,María Medina,Enrico Valdinoci Pdf

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.

Author : Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba
Publisher : Springer Science & Business Media
Page : 547 pages
File Size : 55,7 Mb
Release : 2010-08-16
Category : Mathematics
ISBN : 9783642117060

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Global Analysis of Minimal Surfaces by Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba Pdf

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.

Author : Avy Soffer
Publisher : American Mathematical Soc.
Page : 275 pages
File Size : 48,8 Mb
Release : 2019-03-12
Category : Nonlinear wave equations
ISBN : 9781470441098

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Nonlinear Dispersive Waves and Fluids by Avy Soffer Pdf

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.

Author : Nguyen Loc Hoang
Publisher : World Scientific
Page : 688 pages
File Size : 52,7 Mb
Release : 2017-03-03
Category : Medical
ISBN : 9781786342263

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Multi-wave Medical Imaging: Mathematical Modelling And Imaging Reconstruction by Nguyen Loc Hoang Pdf

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.

Author : Alessandra Lunardi
Publisher : Springer
Page : 199 pages
File Size : 54,6 Mb
Release : 2018-05-05
Category : Mathematics
ISBN : 9788876426384

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Interpolation Theory by Alessandra Lunardi Pdf

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.