Form estimates for the 𝑝(𝑥)-Laplacean

Source abstract

We consider the problem of establishing conditions on p ( x ) p(x) that ensure that the form associated with the p ( x ) p(x) -Laplacean is positive bounded below. It was shown recently by Fan, Zhang and Zhao that—unlike the p = p= constant case—this is not possible if p 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 , p=2, involving arbitrary vector fields. We also examine related results obtained by a variant of Picone Identity arguments. We directly consider problems in Ω ⊂ R n \Omega \subset R^n with n ≥ 1 , 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 ) . p(x). Our basic criteria involve restrictions on p ( x ) p(x) and its gradient.