Quantitative Theory Of Elliptic Homogenization With Robin Boundary Condition On Perforated Domains

This paper is devoted to studying the homogenization problems for second order elliptic operators with Robin boundary condition on perforated domains,i.e.,(?)At first,we show the homogenization theorem.Different from the results and methods developed by Belyaev-Chechkin-Pyatnitski[6]and Chechkin-PiatnitskiShamaev[10],not only do we prove the weak convergence of solution sequences under the energy mode,but also we find the concrete formula on the effective flux quantity associated with the solutions,which is quite crucial in physics and applications.To achieve the goal,we develop an argument by combining two-scale convergences and Tartar's test oscillating function methods,which is new beyond stated result.Based upon the good understanding on flux quantity and WangXu-Zhao's work[21],we obtain a sharp error O(?1/2)in the sense of energy norms,under lower regularity condition on heat source term compared to that assumed in[6,10].Also,it will create a chance to touch optimal convergence rate O(?)in Lp norm(p?2).