Sharp gradient estimates for quasilinear elliptic equations with p(x) growth on nonsmooth domains

Abstract

In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the p(x)-Laplacian on nonsmooth domains and obtain sharp Calderón-Zygmund type estimates in the variable exponent setting. In a recent work of [10], the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above p(x), see (1.3) and (1.4). Here, we bridge this gap to obtain the end point case of the estimates obtained in [10], see (1.5). In order to do this, we have to obtain significantly improved a priori estimates below p(x), which is the main contribution of this paper. We also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature.

End point gradient estimates for quasilinear parabolic equations with variable exponent growth on nonsmooth domains

“…We note that Theorem 5.2 is slightly different than the one proved in [5,Theorem 4.13]. In order to obtain this improvement where the ball B r is the same on both sides of the inequality, we can repeat the arguments in the proof of [5,Theorem 4.13] and combine them with the technical lemma from [29,Lemma 3.4].…”

“…In this section, we shall collect and prove in some cases well known estimates that will be used in subsequent sections. We first recall an integral version of Poincaré's inequality which was proved in [5,Lemma 4.12]:…”

“…Subordinate to the covering G obtained in Lemma A. 5 , we obtain a partition of unity {ψ} ∞ i=1 on R n+1 \ E λ that satisfies the following properties:…”

“…This represented the starting point of obtaining a priori estimates in Sobolev spaces for elliptic and parabolic equations. Since we are interested in Calderón-Zygmund theory for parabolic equations in this paper, we shall discuss the history of the problem only for parabolic equations and refer the reader to [5] and references therein for the elliptic counterpart.…”