HIGHER INTEGRABILITY FOR PARABOLIC SYSTEMS OF p-LAPLACIAN TYPE

it is known that solutions locally belong to a slightly higher Sobolev space than assumed a priori. This self-improving property was first observed by Elcrat and Meyers in [ME] (see also [Gi] and [Str]). Their argument is based on reverse Hölder inequalities and a modification of Gehring’s lemma [Ge], which originally was developed to study the higher integrability of the Jacobian of a quasiconformal mapping. In the elliptic case, higher integrabilty results play a decisive role in studying the regularity of solutions (see [GM] and [Gi]). The purpose of this work is to obtain higher integrablity results in the p-parabolic setting. We prove that the gradient of a weak solution to (1.1) satisfies a reverse Hölder inequality for p > 2n/(n+2). The critical exponent 2n/(n+2) occurs also in parabolic regularity theory (see [D]). We note that reverse Hölder inequalities and the local higher integrability for weak solutions were already proved for p = 2 in [GS] (see also [C]). Our result appears to be new even in the scalar case if p = 2.