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Review of the First Edition: "The monograph is well written and organized and recommended to graduate students and researchers in applied mathematics or engineering." Zentralblatt MATH The primary focus of the book is to explore the asymptotic behavior of problems formulated within cylindrical structures. Various physical applications are discussed, with certain topics such as fluid flows in channels being particularly noteworthy. Additionally, the book delves into the relevance of elasticity in the context of cylindrical bodies. In specific scenarios where the size of the cylinder becomes exceptionally large, the material's behavior is determined solely by its cross-section. The investigation centers around understanding these particular properties. Since the publication of the first edition, several significant advancements have been made, adding depth and interest to the content. Consequently, new sections have been incorporated into the existing edition, complemented by a comprehensive list of references.
Foreword
Introduction
The Dirichlet problem
Periodic problems
Problems in unbounded domains
Rescaling
Eigenvalue problems
Pointwise convergence technique
Calculus of variations problems
Parabolic problems
Strongly nonlinear problems
Higher order operators
Higher order estimates
Global convergence
Correctors for the Dirichlet problem
The Dirichlet problem in some unbounded domains
The linear case
A quasilinear case
Pointwise convergence
Global convergence
The pure Neumann problem
Introduction
The case p = 1
The case of data independent of x1
A case p &gt 1
The case with a lower order term
Periodic problems
A general theory
Some degenerate case
Application to the periodic obstacle problem
The Neumann periodic case
Anisotropic singular perturbation problems
Introduction
Anisotropic singular perturbation problems
Estimates for the rate of convergence
A pure Neumann case
Eigenvalue problems
Introduction
Convergence of the eigenvalues
Convergence of the eigenfunctions
An application
The case of Neumann boundary conditions
Elliptic systems
Abstract formulation
Some applications
The Stokes problem
Introduction and notation
Auxiliary lemmas
Main results
Remark on the Poiseuille flow
Global convergence
Variational inequalities
A simple result
The obstacle problem in unbounded domains
A general framework
Some applications
Calculus of variations
Monotonicity properties
A convergence result
The case of the q-Laplace operator
Some concluding remarks
Dictionary of the main notation
Some existence results
Poincaré’s Inequality
Bibliography
Index