Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with Convex Growth
Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with Convex Growth
复制标题
具有凸增长的非凸无界积分泛函的随机齐次化
DOI：
--
复制
发表时间：
2015
影响因子：
2.5
通讯作者：
A. Gloria
中科院分区：
文献类型：
--
作者：
Mitia Duerinckx;A. Gloria
我们考虑随机积分泛函的均匀化问题。当被积物具有标准生长条件时，定性理论就很好理解了。当涉及到无界泛函时，即当被积函数的定义域不是整个空间并且可能依赖于空间变量时，没有令人满意的理论。在这个贡献中，我们发展了一个完整的具有凸增长的非凸无界泛函的定性随机均匀化理论。我们首先证明了如果被积函数是凸的，并且从下到下有p增长（维数为p b> d），那么无论从上到下的增长条件如何，它都允许均匀化。这个结果，主要依赖于校正的存在性和无穷远处的次线性性，在周期情况下也是新的。在非凸被积的情况下，我们证明了一个类似的齐次化结果成立，前提是非凸被积可以被一个本身允许齐次化的凸被积（其定义域可以依赖于空间变量）的双边估计。这一结果对聚合物物理学中橡胶弹性的严格推导很有意义，因为这涉及到这种无界泛函的随机均匀化。
We consider the well-trodden ground of the problem of the homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the domain of the integrand is not the whole space and may depend on the space-variable, there is no satisfactory theory. In this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth. We first prove that if the integrand is convex and has p-growth from below (with p > d, the dimension), then it admits homogenization regardless of growth conditions from above. This result, that crucially relies on the existence and sublinearity at infinity of correctors, is also new in the periodic case. In the case of nonconvex integrands, we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two-sided estimate by a convex integrand (the domain of which may depend on the space variable) that itself admits homogenization. This result is of interest to the rigorous derivation of rubber elasticity from polymer physics, which involves the stochastic homogenization of such unbounded functionals.