In this paper we study homogenization of a class of control problems in a stationary and ergodic random environment. This problem has been mostly studied in the calculus of variations setting in connection to the homogenization of the Hamilton-Jacobi equations. We extend the result to the control problems with fairly general state dynamics and macroscopically inhomogeneous Lagrangians. Moreover, our approach proves homogenization under weaker growth assumptions on the Lagrangian even in the well-studied calculus of variations setting.
For a homogenization problem associated to a linear elliptic operator, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficients. We also study the convergence rates in the asymptotic almost periodic setting, and we show that the rates of convergence for the zero order approximation, are near optimal. The results obtained constitute a step towards the numerical implementation of results from the deterministic homogenization theory beyond the periodic setting. To illustrate this, numerical simulations based on finite volume method …
Homogenization Results for a Deterministic Multi-domains Periodic Control Problem
我们认为均匀化问题的确定性最优控制的框架下的动态和运行成本是完全不同的两个（或更多）互补域的空间$\R^N$。对于这样的最优控制问题，三个第一作者已经表明，可以定义几个值函数，具体取决于选择是只使用“常规策略”还是使用“奇异策略”。我们在这两种不同的情况下研究均匀化问题。值得指出的是，如果第二种情况可以用通常的偏微分方程方法“á la Lions-Papanicolaou-Varadhan”进行适当的调整来处理，则第一种情况必须用控制方法（动态规划）来处理。
We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case. Specially in the second type, these stochastic solutions are crucial elements in the homogenization analysis. Our homogenization method is based on the notion of stochastic two-scale Young measure, whose existence is established here.
We prove a quantitative rate of homogenization for the G equation in a random environment with finite range of dependence. Using ideas from percolation theory, the proof bootstraps a result of Cardaliaguet and Souganidis, who proved qualitative homogenization in a more general ergodic environment.
In this paper, we systematically study the regularity theory of the linear system of nearly incompressible elasticity. In the setting of stochastic homogenization, we develop new techniques to establish the large-scale estimates of displacement and pressure, which are uniform in both the scale parameter and the incompressibility parameter. In particular, we obtain the boundary estimates in a new class of Lipschitz domains whose boundaries are smooth at large scales and bumpy at small scales.
We study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by …
We prove a stochastic homogenization result for integral functionals defined on finite partitions assuming the surface tension to be stationary and possibly ergodic. We also consider the convergence of boundary value problems when we impose a boundary value just on part of the boundary. As a consequence, we show that if the homogenized surface tension is isotropic, then one can obtain it by a multi-cell problem where Dirichlet boundary conditions are imposed only at the bottom and the top of the cube. We also show that this result fails in the general anisotropic case.
The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodi…
We provide a simple proof of quenched stochastic homogenization for random environments with a mean zero, divergence-free drift under the assumption that the drift admits a stationary $L^d$-integrable stream matrix in $d\geq 3$ or an $L^{2+δ}$-integrable stream matrix in $d=2$. In addition, we prove that the environment almost surely satisfies a large-scale Hölder regularity estimate and first-order Liouville principle.
