标题： 粘性哈密顿-雅可比方程周期均质中的拟凸性丧失 显示英文标题

标题： Loss of quasiconvexity in the periodic homogenization of viscous Hamilton-Jacobi equations

摘要： 我们证明，在任意维数下均匀椭圆哈密顿-雅可比方程的周期同化中，有效哈密顿量不一定继承原哈密顿量在动量变量中的拟凸性。这一观察结果与一阶情况形成鲜明对比，在一阶情况下，已知同化过程会保持拟凸性。我们还证明，拟凸性的损失在某种意义上是普遍的：当空间维数为$1$时，每个凸函数$G$都可以在一个任意小的开区间上进行修改，使得新函数$\tilde{G}$是拟凸的，并且对于某些$1$周期且 Lipschitz 连续的$V$，由具有哈密顿量$H(p,x)=\tilde{G}(p)+V(x)$的均匀椭圆哈密顿-雅可比方程同化得到的有效哈密顿量不是拟凸的。 显示英文摘要

摘要： We show that, in the periodic homogenization of uniformly elliptic Hamilton-Jacobi equations in any dimension, the effective Hamiltonian does not necessarily inherit the quasiconvexity property (in the momentum variables) of the original Hamiltonian. This observation is in sharp contrast with the first order case, where homogenization is known to preserve quasiconvexity. We also show that the loss of quasiconvexity is, in a way, generic: when the spatial dimension is $1$, every convex function $G$ can be modified on an arbitrarily small open interval so that the new function $\tilde{G}$ is quasiconvex and, for some $1$-periodic and Lipschitz continuous $V$, the effective Hamiltonian arising from the homogenization of the uniformly elliptic Hamilton-Jacobi equation with the Hamiltonian $H(p,x)=\tilde{G}(p)+V(x)$ is not quasiconvex.