Differential and Integral Equations

On the $\Gamma$-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part I: The upper bound

Arkady Poliakovsky

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Abstract

In Part I we construct an upper bound, in the spirit of $\Gamma$- $\limsup$, achieved by multidimensional profiles, for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form \begin{equation*} \begin{split} E_{\varepsilon}(v):=\int_\Omega \frac{1}{{\varepsilon}}F\big({\varepsilon}^n\nabla^n v,\dots, {\varepsilon}\nabla v,v\big)\,dx\; \forall v:\Omega\subset \mathbb R ^N\to \mathbb R ^k\; \text{s.t.}\; A\cdot\nabla v=0, \end{split} \end{equation*} where the function $F\geq 0$ and $A: \mathbb R ^{k\times N}\to \mathbb R ^m$ is a prescribed linear operator (for example, $A:\equiv 0$, $A\cdot\nabla v:=\operatorname{curl} v$, and $A\cdot\nabla v=\text{div } v$) which includes, in particular, the problems considered in [17]. This bound is in general sharper then the one obtained in [17]. v$) which includes, in particular, the problems considered in [17]. This bound is in general sharper then the one obtained in [17].

Article information

Source
Differential Integral Equations, Volume 26, Number 9/10 (2013), 1179-1234.

Dates
First available in Project Euclid: 3 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1372858571

Mathematical Reviews number (MathSciNet)
MR3100086

Zentralblatt MATH identifier
1299.35021

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 76D07: Stokes and related (Oseen, etc.) flows 76S05: Flows in porous media; filtration; seepage

Citation

Poliakovsky, Arkady. On the $\Gamma$-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part I: The upper bound. Differential Integral Equations 26 (2013), no. 9/10, 1179--1234. https://projecteuclid.org/euclid.die/1372858571


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