Form estimates for the $p(x)$-Laplacean

We consider the problem of establishing conditions on $p(x)$ that ensure that the form associated with the $p(x)$-Laplacean is positive bounded below. It was shown recently by Fan, Zhang and Zhao that - unlike the $p=$ constant case - this is not possible if $p$ has a strict extrema in the domain. They also considered the closely related problem of eigenvalue existence and estimates. Our main tool is the adaptation of a technique, employed by Protter for $p=2,$ involving arbitrary vector fields. We also examine related results obtained by a variant of Picone Identity arguments. We directly consider problems in $\Omega \subset R^n$ with $n\ge 1,$ and while we focus on Dirichlet boundary conditions we also indicate how our approach can be used in cases of mixed boundary conditions, of unbounded domains and of discontinuous $p(x).$ Our basic criteria involve restrictions on $p(x)$ and its gradient.