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Operator error estimates for homogenization of elliptic systems with periodic coefficients

Presented by: 
T Suslina Saint Petersburg State University
Friday 27th March 2015 - 10:00 to 11:00
INI Seminar Room 1
We study a wide class of matrix elliptic second order differential operators $A_\varepsilon$ in a bounded domain with the Dirichlet or Neumann boundary conditions. The coefficients are assumed to be periodic and depend on $x/\varepsilon$. We are interested in the behavior of the resolvent of $A_\varepsilon$ for small $\varepsilon$. Approximations of this resolvent in the $L_2\to L_2$ and $L_2 \to H^1$ operator norms are obtained. In particular, a sharp order estimate $$ \| (A_\varepsilon - \zeta I)^{-1} - (A^0 - \zeta I)^{-1} \|_{L_2 \to L_2} \le C\varepsilon $$ is proved. Here $A^0$ is the effective operator with constant coefficients.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons