On Homogenization of Locally Periodic Elliptic and Parabolic Operators
Nikita N. Senik · Functional Analysis and Its Applications · 2020
Let Ω be a C1,s bounded domain (s > 1/2) in ℝd, and let $${{\cal A}^\varepsilon } = - {\rm{div}}\;A(x,x/\varepsilon ) abla $$ be a matrix elliptic operator on Ω with Dirichlet boundary condition. We suppose that ε is small and the function A is Lipschitz in the first variable and periodic in the second one, so the coefficients of $${{\cal A}^\varepsilon }$$ are locally periodic. For μ in the resolvent set, we are interested in finding the rates of approximations, as ε → 0, for $${({{\cal A}^\varepsilon } - \mu {\rho ^\varepsilon })^{ - 1}}$$ and $$ abla {({{\cal A}^\varepsilon } - \mu {\rho ^\varepsilon })^{ - 1}}$$ in the operator topology on L2. Here ρε(x)= ρ(x,x/ε) is a positive definite locally periodic function with ρ satisfying the same assumptions as A. Keeping track of the rate dependence on both ε and μ, we then proceed to similar questions for the solution to the initial boundary-value problem $${\rho ^\varepsilon }{\partial _t}{v_\varepsilon } = - {{\cal A}^\varepsilon }{v_\varepsilon }$$ .