The exponential leveling in elliptic singular perturbation problems with complicated attractors (Q804090)

The elliptic differential operator \[ L^{\epsilon}u=\epsilon \sum^{h}_{i,j=1}a_{ij}(x)u_{x_ i}u_{x_ j}+(b(x),\nabla u),\quad \epsilon >0, \] is considered in an open bounded domain \(D\subset {\mathbb{R}}^ n\) with \(C^ 2\)-smooth boundary \(\partial D\). It is proved that the solutions of the equation \(L^{\epsilon}u^{\epsilon}=0\), where \(u^{\epsilon}\in C(\bar D)\cap C^ 2(D)\), satisfy the exponential leveling property \[ \sup_{x,y\in K}| u^{\epsilon}(x)-u^{\epsilon}(y)| \leq \max_{\partial D}| u^{\epsilon}| \exp (-(V(K)-\delta)\epsilon^{-1}) \] for \(0<\epsilon \leq \epsilon (\delta)\), \(\delta >0\) and for any compact \(K\subset D\). Moreover, the constant V(K) is described in terms of the Venttsel'-Frejdlin action functional.