Liouville type results for periodic and almost periodic linear operators

TL;DR

This paper extends Liouville theorems to solutions of periodic and almost periodic linear equations, establishing conditions under which solutions are bounded, periodic, or almost periodic, and exploring the structure of solution spaces.

Contribution

It introduces new Liouville type results for periodic operators, including cases with solutions periodic in one variable and characterizes the solution space dimension under various conditions.

Findings

Bounded solutions are periodic in the same direction as coefficients when certain conditions hold.

The solution space for bounded solutions has at most dimension one under periodicity and non-positivity conditions.

Explicit examples show that almost periodicity cannot be replaced by periodicity without losing the results.

Abstract

We are 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 $c\leq0$ and $a_{ij}, b_i, c, f$ are periodic in the same space/time direction, with the same period, then any bounded solution $u$ of $$\partial_t u-a_{ij}(x,t)\partial_{ij}u-b_i(x,t)\partial_iu-c(x,t)u=f(x,t),\quad x\in\R^N,\ t\in\R,$$ is periodic in that direction. We then derive the following Liouville type result: if $c\leq0, f\equiv0$ 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 $c\leq0$ can be weaken by requiring that the periodic principal eigenvalue 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 $u''+b(x)u'=0$ in $\R$ has dimension 2, and it is generated by the constant solution and a non-almost periodic solution. Next, a sufficient condition for any bounded solution to be almost periodicis derived. We also treat the case of periodic domains under either Dirichlet or Robin boundary conditions.