Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients

Abstract:

Let ${\mathcal O} \subset \mathbb {R}^d$ be a bounded domain of class $C^{2p}$. The object under study is a selfadjoint strongly elliptic operator $A_{D,\varepsilon }$ of order $2p$, $p\geq 2$, in $L_2({\mathcal O};\mathbb {C}^n)$, given by the expression $b(\mathbf D)^* g(\mathbf x/\varepsilon ) b(\mathbf D)$, $\varepsilon >0$, with the Dirichlet boundary conditions. Here $g(\mathbf x)$ is a bounded and positive definite $(m\times m)$-matrix-valued function in $\mathbb {R}^d$, periodic with respect to some lattice; $b(\mathbb {D})=\sum _{|\alpha |=p} b_\alpha \mathbf {D}^\alpha$ is a differential operator of order $p$ with constant coefficients; and the $b_\alpha$ are constant $(m\times n)$-matrices. It is assumed that $m\geq n$ and the symbol $b({\boldsymbol \xi })$ has maximal rank. Approximations are found for the resolvent $(A_{D,\varepsilon } - \zeta I)^{-1}$ in the $L_2({\mathcal O};\mathbb {C}^n)$-operator norm and in the norm of operators acting from $L_2({\mathcal O};\mathbb {C}^n)$ to $H^p({\mathcal O};\mathbb {C}^n)$, with error estimates depending on $\varepsilon$ and $\zeta$.