On the Γ-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part II: The lower bound
On the Γ-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part II: The lower bound
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关于具有非一维最优轮廓的奇异扰动问题的 Γ 极限，第二部分：下界。
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发表时间：
2011
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通讯作者：
A. Poliakovsky
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作者：
A. Poliakovsky
摘要：我们本着 Γ 收敛的精神，为一些一般类别的奇异摄动问题构造下界，无论有或没有规定的微分约束，其形式为 $$E_varepsilon (v): = int_Omega {frac{1}
{变瑞普西隆 }F(变瑞普西隆 ^n
abla ^n v、ldots、varepsilon
abla v,v)dx} for v:Omega 子集 mathbb{R}^N o mathbb{R}^k 使得 A cdot
abla v = 0, $$，其中函数 F 为非负，A: ℝk×N → ℝm 是规定的线性算子（例如，A:≡ 0、A·▿v:=curl v 和 A·▿v = divv）。此外，我们研究了可以轻松证明该下限与[18]中获得的上限一致的情况。特别是，我们找到了一般类别的无微分约束的各向异性问题的 Γ 极限公式（即，在 A: 0 的情况下）。
AbstractWe construct the lower bound, in the spirit of Γ-convergence for some general classes of singular perturbation problems, with or without a prescribed differential constraint, of the form $$E_varepsilon (v): = int_Omega {frac{1}
{varepsilon }F(varepsilon ^n
abla ^n v, ldots ,varepsilon
abla v,v)dx} for v:Omega subset mathbb{R}^N o mathbb{R}^k such that A cdot
abla v = 0, $$ where the function F is nonnegative and A: ℝk×N → ℝm is a prescribed linear operator (for example, A:≡ 0, A · ▿v:= curl v and A · ▿v = divv). Furthermore, we study the cases where we can easily prove that this lower bound coincides with the upper bound obtained in [18]. In particular, we find the formula for the Γ-limit for a general class of anisotropic problems without a differential constraint (i.e., in the case A:≡ 0).