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Un théorème de Liouville pour des équations elliptiques à coefficients périodiques. (A Liouville theorem for elliptic equations with periodic coefficients). (French) Zbl 0691.35022

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

MSC:

35C05 Solutions to PDEs in closed form
35J15 Second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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