Optimisation of space-time periodic eigenvalues

Parabolic periodic eigenvalue problems are important in the study of reaction-diffusion equations, and so is their optimisation with respect to the potential. The main question under consideration is the following: \emph{how to choose $m$ so as to minimise the eigenvalue $\lambda$?} Naturally we would need to specify the proper constraints, but, at a qualitative level, there are two main questions. The first one is the \emph{symmetry} of optimisers: is it true that it always better to replace $m$ with another potential (that satisfies the same constraints) but that is also symmetric in time and in space? The second one, has to do with the \emph{monotonicity} of the optimisers: provided the answer to the first question is positive, is it true that the optimiser is not only symmetric, but also monotonous? Let us emphasise that these questions are answered positively when considering the symmetry and monotonicity with respect to the space variable only.