随机开孔区域中自由间断泛函的随机均匀化
Stochastic homogenisation of free-discontinuity functionals in random perforated domains
摘要：
本文研究了定义在随机穿孔区域上的一族随机自由间断能$E_\varepsilon$在$\varepsilon$趋于零时的渐近行为。泛函$E_\varepsilon$模拟与多孔随机材料的位移相关的能量，该多孔随机材料可以产生裂纹。为了得到能量有界的位移序列的紧性，我们需要克服泛函的不等价性。我们这样做的扩展结果的方式，在假设下，随机穿孔不能来太接近彼此。然后，在两个步骤中获得极限能量。作为第一步，我们适用于一个修改后的，强制性版本的$E_\varepsilon$的自由不连续泛函的随机收敛的一般结果。然后，有效的体积和表面能密度确定通过一个仔细的限制程序。
英文摘要：
In this paper we study the asymptotic behaviour of a family of random free-discontinuity energies $E_\varepsilon$ defined on a randomly perforated domain, as $\varepsilon$ goes to zero. The functionals $E_\varepsilon$ model the energy associated to displacements of porous random materials that can develop cracks. To gain compactness for sequences of displacements with bounded energies, we need to overcome the lack of equi-coerciveness of the functionals. We do so by means of an extension result, under the assumption that the random perforations cannot come too close to one another. The limit energy is then obtained in two steps. As a first step we apply a general result of stochastic convergence of free-discontinuity functionals to a modified, coercive version of $E_\varepsilon$. Then the effective volume and surface energy densities are identified by means of a careful limit procedure.
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