论文标题
差分夹杂物在二维中的定量刚度
Quantitative rigidity of differential inclusions in two dimensions
论文作者
论文摘要
对于任何紧凑的连接的一维submanifold $ k \ subset \ mathbb r^{2 \ times 2} $，它没有等级 - 一个连接并且是椭圆形的，我们证明了定量刚度估计\ [\ inf_ [\ inf_ {m \ in k} du -m |^2 \，dx \ leq c \ int_ {b_1} \ mathrm {dist}^2（du，k）\，dx，\ qquad \ forall u \ in h^1（b_1; \ \ mathbb r^2）。 \]这是一个最佳的概括，对于$ \ mathbb r^{2 \ times 2} $的紧凑连接的子曼属，是Friesecke，James和Müller的著名定量刚度估算值的近似差分估计，以将近似差分纳入$ SO（N）$。证明依赖椭圆子的特殊属性$ k \ subset \ subbb r^{2 \ times 2} $相对于保态 - 单位形式的分解，该分解提供了准确的微分包含$ du \ in k $中的质量椭圆形PDE。我们还举了一个示例，表明没有类似结果可以在$ \ mathbb r^{n \ times n} $中，对于$ n \ geq 3 $。
For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ which has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate \[ \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\,dx \leq C \int_{B_1} \mathrm{dist}^2(Du, K)\, dx, \qquad\forall u\in H^1(B_1;\mathbb R^2). \] This is an optimal generalization, for compact connected submanifolds of $\mathbb R^{2\times 2}$, of the celebrated quantitative rigidity estimate of Friesecke, James and Müller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $K\subset\mathbb R^{2\times 2}$ with respect to conformal-anticonformal decomposition, which provide a quasilinear elliptic PDE satisfied by solutions of the exact differential inclusion $Du\in K$. We also give an example showing that no analogous result can hold true in $\mathbb R^{n\times n}$ for $n\geq 3$.
