论文标题
一个空间维度中一类非凸粘HJ方程的随机均质化
Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension
论文作者
论文摘要
我们证明，在一个空间维度中，在固定的沿着恒定的随机环境中，一类非凸（可能是退化的）粘性汉密尔顿 - 雅各比方程的均匀化。结果涉及表格$ g（p）+v（x，ω）$的汉密尔顿人，其中非线性$ g $是两个或多个凸函数的最低凸功率，具有相同的绝对最小值，并且潜在的$ v $是满足额外缩放山和瓦利条件的界面固定过程。在无粘性情况下，这种情况是在毫无用处地满足的，而在均匀椭圆形的情况下，它等同于A. Yilmaz和O. Zeitouni [31]的原始山丘和山谷条件[31]。我们的方法基于PDE方法，不依赖于解决方案的表示公式。仅使用与适当构建的超级和子解决方案进行比较，我们获得了具有线性初始数据$ x \ mapstoθx$的解决方案的紧密上限和下限。另一个重要成分是P. cardaliaguet和P.E.的一般结果。 Souganidis [13]保证了所有$θ$在与凸的功能相关的有效汉密尔顿人的“平坦零件”外的所有$θ$的存在。我们为这些校正器提供了至关重要的衍生估计，使我们可以将它们用作$ g $的校正器。
We prove homogenization for a class of nonconvex (possibly degenerate) viscous Hamilton-Jacobi equations in stationary ergodic random environments in one space dimension. The results concern Hamiltonians of the form $G(p)+V(x,ω)$, where the nonlinearity $G$ is a minimum of two or more convex functions with the same absolute minimum, and the potential $V$ is a bounded stationary process satisfying an additional scaled hill and valley condition. This condition is trivially satisfied in the inviscid case, while it is equivalent to the original hill and valley condition of A. Yilmaz and O. Zeitouni [31] in the uniformly elliptic case. Our approach is based on PDE methods and does not rely on representation formulas for solutions. Using only comparison with suitably constructed super- and sub- solutions, we obtain tight upper and lower bounds for solutions with linear initial data $x\mapsto θx$. Another important ingredient is a general result of P. Cardaliaguet and P.E. Souganidis [13] which guarantees the existence of sublinear correctors for all $θ$ outside "flat parts" of effective Hamiltonians associated with the convex functions from which $G$ is built. We derive crucial derivative estimates for these correctors which allow us to use them as correctors for $G$.
