Homogenization of Poisson's kernel and applications to boundary control (Q1821943)

We prove an asymptotic expansion for the Poisson kernels \(P_{\epsilon}(x,y)\) of a family of elliptic operators of the form \[ (1)\quad L_{\epsilon}=-(\partial /\partial x^ i)[a^{ij}(x/\epsilon)\partial /\partial x_ j] \] with periodic \(a^{ij}\)'s, in a \(C^{1,\alpha}\) domain. Accordingly, \[ (2)\quad P_{\epsilon}(x,y)=<\partial \Phi ^ *_{\epsilon}/\partial v_{a,\epsilon},\quad n(y)>\partial G_ 0(x,y)/\partial n(y)+R_{\epsilon}(x,y),\quad x\in D,\quad y\in \partial, \] here \(\Phi ^ *_{\epsilon}\) is the solution of \[ (3)\quad L^ *_{\epsilon} \Phi ^ *_{\epsilon} =0\quad in\quad D;\quad \Phi ^ *_{\epsilon}=x\quad on\quad \partial D, \] \(\partial /\partial v_{a,\epsilon}\) is the conormal derivative, \(G_ 0(x,y)\) is the Green's function of the homogenized problem and \(R_{\epsilon}(x,y)\) is a function satisfying \[ (4)\quad | R_{\epsilon}(x,y)| \leq C dist(x,\partial D)/| x-y| ^ n,\quad x\in D,\quad y\in \partial D, \] and \[ (5)\quad \lim _{\epsilon \downarrow 0} \sup _{dist(x,\partial D)>\delta} \sup _{y\in \partial D}| R_{\epsilon}(x,y)| =0, \] for all \(\delta >0\). The expansion (2) is used to identify the limit as \(\epsilon\) \(\to 0\) of solutions to a problem of exact boundary controllability for the hyperbolic operator \(\partial ^ 2_{tt}+L_{\epsilon}\).