Closure and Commutability Results for Γ-Limits and the Geometric Linearization and Homogenization of Multiwell Energy Functionals
Closure and Commutability Results for Γ-Limits and the Geometric Linearization and Homogenization of Multiwell Energy Functionals
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Γ-极限的闭合和可交换性结果以及多孔能量泛函的几何线性化和均匀化
DOI：
10.1137/13093738x
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
B. Schmidt
中科院分区：
文献类型：
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作者：
Martin Jesenko;B. Schmidt
我们考虑一类积分泛函，双索引为 $\varepsilon > 0$ 和 $j \in \mathbb{N} \cup \{\infty\}$，满足一致的G\r 阿丁型不等式。如果这些函数 $\Gamma$-收敛于所有 $j \in \mathbb{N}$ as $\varepsilon \to 0$，那么对于……也是如此 $j = \infty$ 如果密度满足一定的等价条件。这样的话 $\Gamma$-limit for $j = \infty$ 是恢复为极限的 $\Gamma$有限的极限 $j$ as $j \to \infty$。因此，我们得到 $\Gamma$-闭包定理，并进一步找到可交换性的判据 $\Gamma$-限制为 $\varepsilon \to 0$ 和 $j \to \infty$。由于我们从下面的温和增长条件，这个结果不仅为弹性理论中的许多线性化和均匀化结果提供了一个共同的基本原则，而且还允许新的应用，我们通过证明多井能量函数的几何线性化和均匀化交换来举例说明。
We consider a family of integral functionals, doubly indexed by $\varepsilon > 0$ and $j \in \mathbb{N} \cup \{\infty\}$, satisfying a uniform G\r arding-type inequality. If these functionals $\Gamma$-converge for every $j \in \mathbb{N}$ as $\varepsilon \to 0$, then the same holds also for $j = \infty$ if the densities fulfill a certain equivalence condition. In that case the $\Gamma$-limit for $j = \infty$ is recovered as the limit of the $\Gamma$-limits for finite $j$ as $j \to \infty$. We thus obtain a $\Gamma$-closure theorem for such functionals and moreover find criteria for the commutability of the $\Gamma$-limits as $\varepsilon \to 0$ and $j \to \infty$. Due to our mild growth conditions from below this result not only provides a common basic principle for a number of linearization and homogenization results in elasticity theory, but it also allows for new applications which we exemplify by proving that geometric linearization and homogenization of multiwell energy functionals commute.