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Homogenization for non-self-adjoint locally periodic elliptic operators (1703.02023v2)
Published 6 Mar 2017 in math.AP
Abstract: We study the homogenization problem for matrix strongly elliptic operators on $L_2(\mathbb Rd)n$ of the form $\mathcal A\varepsilon=-\operatorname{div}A(x,x/\varepsilon)\nabla$. The function $A$ is Lipschitz in the first variable and periodic in the second. We do not require that $A*=A$, so $\mathcal A\varepsilon$ need not be self-adjoint. In this paper, we provide, for small $\varepsilon$, two terms in the uniform approximation for $(\mathcal A\varepsilon-\mu){-1}$ and a first term in the uniform approximation for $\nabla(\mathcal A\varepsilon-\mu){-1}$. Primary attention is paid to proving sharp-order bounds on the errors of the approximations.