Liouville Type Results for Periodic and Almost Periodic Elliptic Operators

The main feature of this paper concerns extensions of the Liouville theorem to the following class of elliptic equations in non-divergence form: aij(x)∂iju + bi(x)∂iu + c(x)u = 0 in R N , with c ≤ 0. We show that the Liouville property holds (that is, the space of bounded solutions has at most dimension one) if the coefficients aij , bi and c are periodic, with the same period, and it does not hold in general if the coefficients are only almost periodic. The Liouville property for periodic operators was already proved in [11], using homogenization technics and Floquet theory. Here, we follow a completely different and more direct approach, deriving the Liouville property from the following result, which is of independent interest: any bounded solution of −aij(x)∂iju − bi(x)∂iu − c(x)u = f(x) in R N , with c ≤ 0 and aij , bi, c, f periodic in the same variable, with the same period, is periodic in that variable. In contrast, bounded solutions of almost periodic equations with nonpositive zero order coefficient are not necessarily almost periodic, as we explicitly show with a counterexample. We further consider the problem of almost periodicity of bounded solutions of equations with periodic coefficients and almost periodic term f . Finally, we establish analogous results to those mentioned above for either Dirichlet or oblique derivative problems in general unbounded periodic domains.