Rate of decrease at infinity of solutions of partial differential equations with periodic coefficients (Q1063787)

Let \(L=L(x,D)=\sum_{| \alpha | \leq m}a_{\alpha}(x)D^{\alpha}\) be a linear elliptic differential operator in \(R^ n\), whose coefficients are infinitely differentiable functions, having periods \(\ell_ i\) in \(x_ i\), \(i=1,2,...,n\); N is a set of quasi-impulses corresponding to the non-vanishing Blochovsky solutions of the equation \(Lu=0\). Let the cone K be normal with respect to N. Theorem 1 is central in the paper: Let \(L=L(x,D)\) be as above, \(2\pi\) periodic in every variable \(x_ i\), \(i=1,2,...,n\). Let, moreover, \(u\in {\mathcal S}'(R^ n)\) and satisfy the equation \(Lu=0\). Assume that there holds \(N\neq R^ n.\) Then, if the codimension of the set N in \(R^ n\) is K and \[ \underline{\lim}_{R\to \infty}R^{-k}\int_{K_{R,2}}| u(x)|^ 2dx\neq 0 \] there follows \(u(x)=0\).