Liouville type results for periodic and almost periodic linear operators

Abstract

This paper is concerned with some extensions of the classical Liouville theorem for bounded harmonic functions to solutions of more general equations. 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 $a_{ij}$, $b_i$, $c$, $f$ are periodic in the same space direction or in time, with the same period, then any bounded solution $u$ of

\is periodic in that direction or in time. We then derive the following Liouville type result: if $c⩽0$, $f\equiv 0$ and $a_{ij}$, $b_i$, $c$ are periodic in all the space/time variables, with the same periods, then the space of bounded solutions of the above equation has at most dimension one. In the case of the equation ${\partial }_{t}u-Lu=f(x,t)$, with $L$ periodic elliptic operator independent of $t$, the hypothesis $c⩽0$ can be weakened by requiring that the periodic principal eigenvalue ${\lambda }_p$ of $-L$ is nonnegative. Instead, the periodicity assumption cannot be relaxed, because we explicitly exhibit an almost periodic function $b$ such that the space of bounded solutions of $u^{\prime \prime }+b(x)u^{\prime }=0$ in $\mathbb{R}$ has dimension 2, and it is generated by the constant solution and a non-almost periodic solution.

The above counterexample leads us to consider the following problem: under which conditions are bounded solutions necessarily almost periodic? We show that a sufficient condition in the case of the equation ${\partial }_{t}u-Lu=f(x,t)$ is: $f$ is almost periodic and $L$ is periodic with ${\lambda }_p⩾0$.

Finally, we consider problems in general periodic domains under either Dirichlet or Robin boundary conditions. We prove analogous properties as in the whole space, together with some existence and uniqueness results for entire solutions.