The Effect Of A Singular Perturbation To A 1 D Non Convex Variational Problem
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The Effect Of A Singular Perturbation To A 1 D Non Convex Variational Problem
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Author : Markus Lilli
language : en
Publisher: Logos Verlag Berlin
Release Date : 2005
The Effect Of A Singular Perturbation To A 1 D Non Convex Variational Problem written by Markus Lilli and has been published by Logos Verlag Berlin this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Nichtkonvexes Variationsproblem - Singuläre Störung - Euler-Lagrange-Gleichung categories.
Nonconvex variational problems are of importance in modeling problems of microstructures and elasticity. In this book, we consider a $1$-d nonconvex problem and we prove existence of solutions of the corresponding non-elliptic Euler-Lagrange equation by considering the Euler-Lagrange equation of the singular perturbed variational problem and passing to the limit. Under general assumptions on the potential we prove existence of Young-measure solutions. More restrictive conditions on the potential yield classical solutions via a topological method. The singular perturbed problem, which is also of interest for physicists due to the higher gradient surface-energy, is discussed in big detail.
Commutability Of Gamma Limits In Problems With Multiple Scales
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Author : Martin Jesenko
language : en
Publisher: Logos Verlag Berlin GmbH
Release Date : 2017-05-15
Commutability Of Gamma Limits In Problems With Multiple Scales written by Martin Jesenko and has been published by Logos Verlag Berlin GmbH this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-05-15 with Mathematics categories.
In the calculus of variations, the goal is to explore extrema of a given integral functional. From origins of the problem, it might be expected that the functional can be adequately simplified by neglecting some small quantities. A way to rigorously justify such an approximation is the Γ-convergence that ensures convergence of corresponding (global) extrema. The main motivation of this work is to investigate properties of doubly indexed integral functionals that Γ-converge for one index fixed. In other words, for two possible approximations we would like to determine whether we may perform them consecutively and if they commute. Our examples are taken from material science with homogenization being one of these two processes. In the first part we are considering a setting related to the elastic regime. However, our assumptions are fairly general and allow for applications in different areas. The second part is devoted to problems in the Hencky plasticity. They are considerably different due to special growth properties of the density.
Global Regularity And Uniqueness Of Solutions In A Surface Growth Model Using Rigorous A Posteriori Methods
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Author : Christian Nolde
language : en
Publisher: Logos Verlag Berlin GmbH
Release Date : 2017-04-20
Global Regularity And Uniqueness Of Solutions In A Surface Growth Model Using Rigorous A Posteriori Methods written by Christian Nolde and has been published by Logos Verlag Berlin GmbH this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-04-20 with Mathematics categories.
The use of rigorous numerical methods to approach problems which can not be solved using standard methods (yet) has increased signifiantly in recent years. In this book, riogorous a-posteriori methods are used to study the time evolution of a surface growth model, given by a fourth order semi-linear parabolic partial differential equation, where standard methods fail to verify global uniqueness and smoothness of solutions. Based on an arbitrary numerical approximation, a-posteriori error-analysis is applied in order to prevent a blow up analytically. This is a method that in a similar way also applies to the three dimensional Navier-Stokes equations. The main idea consists of energy-estimates for the error between solution and approximation that yields a scalar differential equation controlling the norm of the error with coefficients depending solely on the numerical data. This allows the solution of the differential equation to be bounded using only numerical data. A key technical tool is a rigorous eigenvalue bound for the nonlinear operator linearized around the numerical approximation. The presented method succeeds to show global uniqueness for relatively large initial conditions, which is demonstrated in many numerical examples.
The Mechanics And Thermodynamics Of Continuous Media
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Author : Miroslav Silhavy
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-27
The Mechanics And Thermodynamics Of Continuous Media written by Miroslav Silhavy and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-27 with Science categories.
From the reviews: "The book is excellent, and covers a very broad area (usually treated as separate topics) from a unified perspective. [...] It will be very useful for both mathematicians and physicists." EMS Newsletter
Nonlinear Analysis And Optimization
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Author : Boris S. Mordukhovich
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-02-26
Nonlinear Analysis And Optimization written by Boris S. Mordukhovich and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-02-26 with Game theory, economics, social and behavioral sciences categories.
This volume contains the proceedings of the IMU/AMS Special Session on Nonlinear Analysis and Optimization, held from June 16-19, 2014, at the Second Joint International Meeting of the Israel Mathematical Union (IMU) and the American Mathematical Society (AMS), Bar-Ilan and Tel-Aviv Universities, Israel, and the Workshop on Nonlinear Analysis and Optimization, held on June 12, 2014, at the Technion-Israel Institute of Technology. The papers in this volume cover many different topics in Nonlinear Analysis and Optimization, including: Taylor domination property for analytic functions in the complex disk, mappings with upper integral bounds for p -moduli, multiple Fourier transforms and trigonometric series in line with Hardy's variation, finite-parameter feedback control for stabilizing damped nonlinear wave equations, implicit Euler approximation and optimization of one-sided Lipschitz differential inclusions, Bolza variational problems with extended-valued integrands on large intervals, first order singular variational problem with nonconvex cost, gradient and extragradient methods for the elasticity imaging inverse problem, discrete approximations of the entropy functional for probability measures on the plane, optimal irrigation scheduling for wheat production, existence of a fixed point of nonexpansive mappings in uniformly convex Banach spaces, strong convergence properties of m-accretive bounded operators, the Reich-Simons convex analytic inequality, nonlinear input-output equilibrium, differential linear-quadratic Nash games with mixed state-control constraints, and excessive revenue models of competitive markets.
Mathematical Problems In Image Processing
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Author : Gilles Aubert
language : en
Publisher: Springer Science & Business Media
Release Date : 2001-11-09
Mathematical Problems In Image Processing written by Gilles Aubert and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-11-09 with Mathematics categories.
Partial differential equations and variational methods were introduced into image processing about 15 years ago, and intensive research has been carried out since then. The main goal of this work is to present the variety of image analysis applications and the precise mathematics involved. It is intended for two audiences. The first is the mathematical community, to show the contribution of mathematics to this domain and to highlight some unresolved theoretical questions. The second is the computer vision community, to present a clear, self-contained, and global overview of the mathematics involved in image processing problems. The book is divided into five main parts. Chapter 1 is a detailed overview. Chapter 2 describes and illustrates most of the mathematical notions found throughout the work. Chapters 3 and 4 examine how PDEs and variational methods can be successfully applied in image restoration and segmentation processes. Chapter 5, which is more applied, describes some challenging computer vision problems, such as sequence analysis or classification. This book will be useful to researchers and graduate students in mathematics and computer vision.
Existence And Regularity Of Minimizers For A One Dimensional Elasticity Problem Without Convexity
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Author : María Mercedes Franco
language : en
Publisher:
Release Date : 2005
Existence And Regularity Of Minimizers For A One Dimensional Elasticity Problem Without Convexity written by María Mercedes Franco and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with categories.
Differential Geometry And Continuum Mechanics
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Author : Gui-Qiang G. Chen
language : en
Publisher: Springer
Release Date : 2015-08-11
Differential Geometry And Continuum Mechanics written by Gui-Qiang G. Chen and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-08-11 with Mathematics categories.
This book examines the exciting interface between differential geometry and continuum mechanics, now recognised as being of increasing technological significance. Topics discussed include isometric embeddings in differential geometry and the relation with microstructure in nonlinear elasticity, the use of manifolds in the description of microstructure in continuum mechanics, experimental measurement of microstructure, defects, dislocations, surface energies, and nematic liquid crystals. Compensated compactness in partial differential equations is also treated. The volume is intended for specialists and non-specialists in pure and applied geometry, continuum mechanics, theoretical physics, materials and engineering sciences, and partial differential equations. It will also be of interest to postdoctoral scientists and advanced postgraduate research students. These proceedings include revised written versions of the majority of papers presented by leading experts at the ICMS Edinburgh Workshop on Differential Geometry and Continuum Mechanics held in June 2013. All papers have been peer reviewed.
Research In Mathematics Of Materials Science
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Author : Malena I. Español
language : en
Publisher: Springer Nature
Release Date : 2022-09-27
Research In Mathematics Of Materials Science written by Malena I. Español and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-09-27 with Mathematics categories.
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.
Gamma Convergence For Beginners
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Author : Andrea Braides
language : en
Publisher: Clarendon Press
Release Date : 2002-07-25
Gamma Convergence For Beginners written by Andrea Braides and has been published by Clarendon Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002-07-25 with Mathematics categories.
The theory of Gamma-convergence is commonly recognized as an ideal and flexible tool for the description of the asymptotic behaviour of variational problems. Its applications range from the mathematical analysis of composites to the theory of phase transitions, from Image Processing to Fracture Mechanics. This text, written by an expert in the field, provides a brief and simple introduction to this subject, based on the treatment of a series of fundamental problems that illustrate the main features and techniques of Gamma-convergence and at the same time provide a stimulating starting point for further studies. The main part is set in a one-dimensional framework that highlights the main issues of Gamma-convergence without the burden of higher-dimensional technicalities. The text deals in sequence with increasingly complex problems, first treating integral functionals, then homogenisation, segmentation problems, phase transitions, free-discontinuity problems and their discrete and continuous approximation, making stimulating connections among those problems and with applications. The final part is devoted to an introduction to higher-dimensional problems, where more technical tools are usually needed, but the main techniques of Gamma-convergence illustrated in the previous section may be applied unchanged. The book and its structure originate from the author's experience in teaching courses on this subject to students at PhD level in all fields of Applied Analysis, and from the interaction with many specialists in Mechanics and Computer Vision, which have helped in making the text addressed also to a non-mathematical audience. The material of the book is almost self-contained, requiring only some basic notion of Measure Theory and Functional Analysis.