EXPLORING THE EQUIVALENCE BETWEEN THE POINCARE PROPERTY OF ORDER P AND THE P-NEUMANN PROPERTY IN THE VARIABLE EXPONENT SETTING

In [5], it was shown under weak assumptions on a matrix function Q that the Poincaré property of order p is equivalent to the p-Neumann property, where 1 < p < \infty is a constant exponent. We attempt to translate this result into into the variable exponent setting by replacing p with a function p(\cdot). To do so, we translate the Banach function spaces L^p, and \mathscr{L}_Q^p , and the Sobolev space H_Q^{1;p} into their variable versions, L^{p(\cdot)}, \mathscr{L}_Q^{p(\cdot)}, and H_Q^{1;p(\cdot)}, and investigate whether the necessary properties of these spaces still hold. We then attempt to replicate the arguments in [5], and conclude that some arguments do not translate well.