论文标题
椭圆形均质化，几乎翻译不变系数
Elliptic homogenization with almost translation-invariant coefficients
论文作者
论文摘要
我们考虑二阶椭圆方程$ - \ peripatorName {div} \ left（a（./ \ varepsilon）\ nabla u^{\ varepsilon} \ right）= f $的均化问题。该几何形状的特征是系数$ a $的特定离散梯度，该梯度属于lebesgue space $ l^p（\ mathbb {r}^d）$ for $ p \ in [1，+\ infty [$。当$ p <d $时，我们建立了gagliardo-nirenberg-sobolev不平等的离散改编，以表明系数$ a $实际上属于特定类别受到本地缺陷扰动的周期性系数。接下来，我们证明了校正器的存在，并确定了$ u^{\ varepsilon} $的均质限制。当$ p \ geq d $时，我们展示了可接受的系数$ a $，因此$ u^{\ varepsilon} $具有不同的子序列，这些子序列在$ l^2 $中收敛到不同的限制。
We consider an homogenization problem for the second order elliptic equation $-\operatorname{div}\left(a(./\varepsilon) \nabla u^{\varepsilon} \right)=f$ when the coefficient $a$ is almost translation-invariant at infinity and models a geometry close to a periodic geometry. This geometry is characterized by a particular discrete gradient of the coefficient $a$ that belongs to a Lebesgue space $L^p(\mathbb{R}^d)$ for $p\in[1,+\infty[$. When $p<d$, we establish a discrete adaptation of the Gagliardo-Nirenberg-Sobolev inequality in order to show that the coefficient $a$ actually belongs to a certain class of periodic coefficients perturbed by a local defect. We next prove the existence of a corrector and we identify the homogenized limit of $u^{\varepsilon}$. When $p\geq d$, we exhibit admissible coefficients $a$ such that $u^{\varepsilon}$ possesses different subsequences that converge to different limits in $L^2$.
