Liouville type results for periodic and almost periodic linear operators

Rossi This paper is concerned with some extensions of the classical Liouville theorem for bounded harmonic functions to solutions of more general equa-tions. We deal with entire solutions of periodic and almost periodic parabolic equations including the elliptic framework as a particular case. We derive a Liouville type result for periodic operators as a consequence of a result for operators periodic in just one variable, which is new even in the elliptic case. More precisely, we show that if c ≤ 0 and aij, bi, c, f are periodic in the same space direction or in time, with the same period, then any bounded solution u of ∂tu − aij(x, t)∂iju − bi(x, t)∂iu − c(x, t)u = f(x, t), x ∈ R