On the $Gamma$-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part I: The upper bound
On the $Gamma$-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part I: The upper bound
复制标题
具有非一维最优轮廓的奇异扰动问题的 $Gamma$ 极限。
DOI：
--
复制
发表时间：
2011
影响因子：
1.4
通讯作者：
A. Poliakovsky
中科院分区：
文献类型：
--
作者：
A. Poliakovsky
在第一部分中，我们本着$Gamma$-$limsup$的精神，构造了一些一般类型的奇异摄动问题的上界，无论有没有规定的微分约束，其形式为$$E_e(V)：=int_Omega frac{1}{e}FBig(e^n
Abla^n v，...，e
Abla v，vBig)dxquad ext{for}v：OmeasubsetR^N或^k ext{从而}Acdo
ABLA v=0，$$其中函数$Fgeq 0$和$A：r^{k imes N}或^m$是规定的线性运算符(例如，$A：EQUIV 0$，$ACTDOT
Abla v：=ext{curl}v$和$Acdo
Abla v=ext{div}，v$)，特别包括在[27]中考虑的问题。一般来说，这个界限比[27]中得到的界限更尖锐。
In Part I we construct the 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 $$E_e(v):=int_Omega frac{1}{e}FBig(e^n
abla^n v,...,e
abla v,vBig)dxquad ext{for} v:OmegasubsetR^N oR^k ext{such that} Acdot
abla v=0,$$ where the function $Fgeq 0$ and $A:R^{k imes N} oR^m$ is a prescribed linear operator (for example, $A:equiv 0$, $Acdot
abla v:= ext{curl}v$ and $Acdot
abla v= ext{div},v$) which includes, in particular, the problems considered in [27]. This bound is in general sharper then one obtained in [27].