Author: Hauke HankePublisher:
ISBN:
Category :
Languages : en
Pages : 28
Book Description
Homogenization of Elliptic Systems with Non-periodic, State Dependent Coefficients
Author: Hauke HankePeriodic Homogenization of Elliptic Systems
Author: Zhongwei ShenPublisher: Springer
ISBN: 3319912143
Category : Mathematics
Languages : en
Pages : 291
Book Description
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.
Elliptic Systems of Phase Transition Type
Author: Nicholas D. AlikakosPublisher: Springer
ISBN: 3319905724
Category : Mathematics
Languages : en
Pages : 343
Book Description
This book focuses on the vector Allen-Cahn equation, which models coexistence of three or more phases and is related to Plateau complexes – non-orientable objects with a stratified structure. The minimal solutions of the vector equation exhibit an analogous structure not present in the scalar Allen-Cahn equation, which models coexistence of two phases and is related to minimal surfaces. The 1978 De Giorgi conjecture for the scalar problem was settled in a series of papers: Ghoussoub and Gui (2d), Ambrosio and Cabré (3d), Savin (up to 8d), and del Pino, Kowalczyk and Wei (counterexample for 9d and above). This book extends, in various ways, the Caffarelli-Córdoba density estimates that played a major role in Savin's proof. It also introduces an alternative method for obtaining pointwise estimates. Key features and topics of this self-contained, systematic exposition include: • Resolution of the structure of minimal solutions in the equivariant class, (a) for general point groups, and (b) for general discrete reflection groups, thus establishing the existence of previously unknown lattice solutions. • Preliminary material beginning with the stress-energy tensor, via which monotonicity formulas, and Hamiltonian and Pohozaev identities are developed, including a self-contained exposition of the existence of standing and traveling waves. • Tools that allow the derivation of general properties of minimizers, without any assumptions of symmetry, such as a maximum principle or density and pointwise estimates. • Application of the general tools to equivariant solutions rendering exponential estimates, rigidity theorems and stratification results. This monograph is addressed to readers, beginning from the graduate level, with an interest in any of the following: differential equations – ordinary or partial; nonlinear analysis; the calculus of variations; the relationship of minimal surfaces to diffuse interfaces; or the applied mathematics of materials science.
Some Linear Elliptic Systems of Partial Differential Equation with Doubly Periodic Coefficients
Author: Mrs. Sondra Osher JaffePublisher:
ISBN:
Category :
Languages : en
Pages : 144
Book Description
Elliptic Systems in the Plane
Author: Wolfgang L. WendlandPublisher: Pitman Publishing
ISBN:
Category : Mathematics
Languages : en
Pages : 424
Book Description
An Introduction to Stochastic Differential Equations with Reflection
Author: Andrey PilipenkoPublisher: Universitätsverlag Potsdam
ISBN: 3869562978
Category :
Languages : en
Pages : 90
Book Description
Second Order Elliptic Equations and Elliptic Systems
Author: Yazhe ChenPublisher: Amer Mathematical Society
ISBN: 9780821809709
Category : Mathematics
Languages : en
Pages : 246
Book Description
The first part of this book presents a complete introduction of various kinds of a priori estimate methods for the Dirichlet problem of second order elliptic partial differential equations are completely introduced. In the second part, the existence and regularity theory of the Dirichlet problem for linear and nonlinear second order elliptic partial differential systems are introduced. The book features appropriate materials and is an excellent textbook for graduate students.
Corrector Estimates for Elliptic Systems with Random Periodic Coefficients
Author: Peter BellaHomogenization
Author: Gregori A. ChechkinPublisher: American Mathematical Soc.
ISBN: 9780821889701
Category : Mathematics
Languages : en
Pages : 256
Book Description
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.
Quantitative Stochastic Homogenization and Large-Scale Regularity
Author: Scott ArmstrongPublisher: Springer
ISBN: 3030155455
Category : Mathematics
Languages : en
Pages : 518
Book Description
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.