Avellaneda, Marco; Lin, Fang-Hua Un théorème de Liouville pour des équations elliptiques à coefficients périodiques. (A Liouville theorem for elliptic equations with periodic coefficients). (French) Zbl 0691.35022 C. R. Acad. Sci., Paris, Sér. I 309, No. 5, 245-250 (1989). Consider an elliptic divergence form operator \(L=\nabla \cdot a(x)\cdot \nabla\), where \(a(x)=(a_{ij}(x))\), is a Lipschitz continuous, positive definite, periodic, second-order tensor. We classify all global solutions u, satisfying \(Lu(x)=0\), having polynomial growth. This is done by constructing a class of solutions \(\{Q_{n,j}(x)\}\), \(j\leq v_ n\), \(n\geq 0\) satisfying \(| Q_{n,j}(x)| \leq C_{n,j}| x|^ n\), such that every solution u(x), with \(Lu(x)=0\) and \(| u| (x)\leq C| x|^ N\) can be expressed in the form \(u(x)=\sum_{| n| \leq N}\sum_{j\leq v_ n}C_{n,j}Q_{n,j}(x)\). Reviewer: M.Avellaneda Cited in 3 ReviewsCited in 32 Documents MSC: 35C05 Solutions to PDEs in closed form 35J15 Second-order elliptic equations 35B40 Asymptotic behavior of solutions to PDEs Keywords:Liouville theorem; homogenization; harmonic polynomials PDFBibTeX XMLCite \textit{M. Avellaneda} and \textit{F.-H. Lin}, C. R. Acad. Sci., Paris, Sér. I 309, No. 5, 245--250 (1989; Zbl 0691.35022)