Download or read book Periodic Homogenization of Elliptic Systems written by Zhongwei Shen and published by Springer. This book was released on 2018-09-04 with total page 291 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book An Introduction to Homogenization written by Doïna Cioranescu and published by Oxford University Press on Demand. This book was released on 1999 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: Composite materials are widely used in industry: well-known examples of this are the superconducting multi-filamentary composites which are used in the composition of optical fibres. Such materials are complicated to model, as different points in the material will have different properties. The mathematical theory of homogenization is designed to deal with this problem, and hence is used to model the behaviour of these important materials. This book provides a self-contained and authoritative introduction to the subject for graduates and researchers in the field.
Download or read book Homogenization of Differential Operators and Integral Functionals written by V.V. Jikov and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 583 pages. Available in PDF, EPUB and Kindle. Book excerpt: It was mainly during the last two decades that the theory of homogenization or averaging of partial differential equations took shape as a distinct mathe matical discipline. This theory has a lot of important applications in mechanics of composite and perforated materials, filtration, disperse media, and in many other branches of physics, mechanics and modern technology. There is a vast literature on the subject. The term averaging has been usually associated with the methods of non linear mechanics and ordinary differential equations developed in the works of Poincare, Van Der Pol, Krylov, Bogoliubov, etc. For a long time, after the works of Maxwell and Rayleigh, homogeniza tion problems for· partial differential equations were being mostly considered by specialists in physics and mechanics, and were staying beyond the scope of mathematicians. A great deal of attention was given to the so called disperse media, which, in the simplest case, are two-phase media formed by the main homogeneous material containing small foreign particles (grains, inclusions). Such two-phase bodies, whose size is considerably larger than that of each sep arate inclusion, have been discovered to possess stable physical properties (such as heat transfer, electric conductivity, etc.) which differ from those of the con stituent phases. For this reason, the word homogenized, or effective, is used in relation to these characteristics. An enormous number of results, approximation formulas, and estimates have been obtained in connection with such problems as electromagnetic wave scattering on small particles, effective heat transfer in two-phase media, etc.
Download or read book Harmonic Analysis and Applications written by Carlos E. Kenig and published by American Mathematical Soc.. This book was released on 2020-12-14 with total page 345 pages. Available in PDF, EPUB and Kindle. Book excerpt: The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments.
Download or read book Elliptic Equations An Introductory Course written by Michel Chipot and published by Springer Science & Business Media. This book was released on 2009-02-19 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and refinements. Apart from the basic theory of equations in divergence form it includes subjects such as singular perturbation problems, homogenization, computations, asymptotic behaviour of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes system, p-Laplace equation. Just a minimum on Sobolev spaces has been introduced, and work or integration on the boundary has been carefully avoided to keep the reader's attention on the beauty and variety of these issues. The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original and have not been published elsewhere. The book will be of interest to graduate students and faculty members specializing in partial differential equations.
Download or read book The Periodic Unfolding Method written by Doina Cioranescu and published by Springer. This book was released on 2018-11-03 with total page 515 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book on the subject of the periodic unfolding method (originally called "éclatement périodique" in French), which was originally developed to clarify and simplify many questions arising in the homogenization of PDE's. It has since led to the solution of some open problems. Written by the three mathematicians who developed the method, the book presents both the theory as well as numerous examples of applications for partial differential problems with rapidly oscillating coefficients: in fixed domains (Part I), in periodically perforated domains (Part II), and in domains with small holes generating a strange term (Part IV). The method applies to the case of multiple microscopic scales (with finitely many distinct scales) which is connected to partial unfolding (also useful for evolution problems). This is discussed in the framework of oscillating boundaries (Part III). A detailed example of its application to linear elasticity is presented in the case of thin elastic plates (Part V). Lastly, a complete determination of correctors for the model problem in Part I is obtained (Part VI). This book can be used as a graduate textbook to introduce the theory of homogenization of partial differential problems, and is also a must for researchers interested in this field.
Download or read book Asymptotic Analysis for Periodic Structures written by Alain Bensoussan and published by American Mathematical Soc.. This book was released on 2011-10-26 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence methods. Before this book, multiple scale methods were primarily used for non-linear oscillation problems in the applied mathematics community, not for analyzing spatial oscillations as in homogenization. In the current printing a number of minor corrections have been made, and the bibliography was significantly expanded to include some of the most important recent references. This book gives systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate. The book continues to be interesting and useful to readers of different backgrounds, both from pure and applied mathematics, because of its informal style of introducing the multiple scale methodology and the detailed proofs.
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.
Download or read book Partial Differential Equations in Anisotropic Musielak Orlicz Spaces written by Iwona Chlebicka and published by Springer Nature. This book was released on 2021-11-01 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a detailed study of nonlinear partial differential equations satisfying certain nonstandard growth conditions which simultaneously extend polynomial, inhomogeneous and fully anisotropic growth. The common property of the many different kinds of equations considered is that the growth conditions of the highest order operators lead to a formulation of the equations in Musielak–Orlicz spaces. This high level of generality, understood as full anisotropy and inhomogeneity, requires new proof concepts and a generalization of the formalism, calling for an extended functional analytic framework. This theory is established in the first part of the book, which serves as an introduction to the subject, but is also an important ingredient of the whole story. The second part uses these theoretical tools for various types of PDEs, including abstract and parabolic equations but also PDEs arising from fluid and solid mechanics. For connoisseurs, there is a short chapter on homogenization of elliptic PDEs. The book will be of interest to researchers working in PDEs and in functional analysis.
Download or read book Mathematical Problems in Elasticity and Homogenization written by O.A. Oleinik and published by Elsevier. This book was released on 1992-11-02 with total page 397 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is based on research undertaken by the authors during the last ten years. The main part of the work deals with homogenization problems in elasticity as well as some mathematical problems related to composite and perforated elastic materials. This study of processes in strongly non-homogeneous media brings forth a large number of purely mathematical problems which are very important for applications. Although the methods suggested deal with stationary problems, some of them can be extended to non-stationary equations. With the exception of some well-known facts from functional analysis and the theory of partial differential equations, all results in this book are given detailed mathematical proof. It is expected that the results and methods presented in this book will promote further investigation of mathematical models for processes in composite and perforated media, heat-transfer, energy transfer by radiation, processes of diffusion and filtration in porous media, and that they will stimulate research in other problems of mathematical physics and the theory of partial differential equations.
Download or read book Quantitative Stochastic Homogenization and Large Scale Regularity written by Scott Armstrong and published by Springer. This book was released on 2019-05-09 with total page 518 pages. Available in PDF, EPUB and Kindle. Book excerpt: The focus of this book is the large-scale statistical behavior of solutions of divergence-form elliptic equations with random coefficients, which is closely related to the long-time asymptotics of reversible diffusions in random media and other basic models of statistical physics. Of particular interest is the quantification of the rate at which solutions converge to those of the limiting, homogenized equation in the regime of large scale separation, and the description of their fluctuations around this limit. This self-contained presentation gives a complete account of the essential ideas and fundamental results of this new theory of quantitative stochastic homogenization, including the latest research on the topic, and is supplemented with many new results. The book serves as an introduction to the subject for advanced graduate students and researchers working in partial differential equations, statistical physics, probability and related fields, as well as a comprehensive reference for experts in homogenization. Being the first text concerned primarily with stochastic (as opposed to periodic) homogenization and which focuses on quantitative results, its perspective and approach are entirely different from other books in the literature.
Download or read book Homogenization of Multiple Integrals written by Andrea Braides and published by Oxford University Press. This book was released on 1998 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to the mathematical theory of the homogenization of multiple integrals, this book describes the overall properties of such functionals with various applications ranging from cellular elastic materials to Riemannian metrics.
Download or read book Mathematical and Statistical Methods for Multistatic Imaging written by Habib Ammari and published by Springer. This book was released on 2013-11-29 with total page 361 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers recent mathematical, numerical, and statistical approaches for multistatic imaging of targets with waves at single or multiple frequencies. The waves can be acoustic, elastic or electromagnetic. They are generated by point sources on a transmitter array and measured on a receiver array. An important problem in multistatic imaging is to quantify and understand the trade-offs between data size, computational complexity, signal-to-noise ratio, and resolution. Another fundamental problem is to have a shape representation well suited to solving target imaging problems from multistatic data. In this book the trade-off between resolution and stability when the data are noisy is addressed. Efficient imaging algorithms are provided and their resolution and stability with respect to noise in the measurements analyzed. It also shows that high-order polarization tensors provide an accurate representation of the target. Moreover, a dictionary-matching technique based on new invariants for the generalized polarization tensors is introduced. Matlab codes for the main algorithms described in this book are provided. Numerical illustrations using these codes in order to highlight the performance and show the limitations of numerical approaches for multistatic imaging are presented.
Download or read book Selected Topics in Almost Periodicity written by Marko Kostić and published by Walter de Gruyter GmbH & Co KG. This book was released on 2021-11-22 with total page 606 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covers uniformly recurrent solutions and c-almost periodic solutions of abstract Volterra integro-differential equations as well as various generalizations of almost periodic functions in Lebesgue spaces with variable coefficients. Treats multi-dimensional almost periodic type functions and their generalizations in adequate detail.
Download or read book Homogenization written by Gregori A. Chechkin and published by American Mathematical Soc.. This book was released on with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on both classical results of homogenization theory and modern techniques developed over the past decade. The powerful techniques in partial differential equations are illustrated with many exercises and examples to enhance understanding of the material. Several of the modern topics that are presented have not previously appeared in any monograph.
Download or read book 75 Years of Mathematics of Computation written by Susanne C. Brenner and published by American Mathematical Soc.. This book was released on 2020-07-29 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Gratings Theory and Numeric Applications written by and published by Popov, Institut Fresnel. This book was released on with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt:
