Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Regularity for subquadratic parabolic systems: higher integrability and dimension estimates

Christoph Schevena1

a1 Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Bismarckstrasse 1½, 91054 Erlangen, Germany (scheven@mi.uni-erlangen.de)

Abstract

We consider weak solutions of parabolic systems of the type

$$
\partial_tu-\mathrm{div} b(x,t,Du)=G\quad\text{on }\varOmega\times(-T,0)\subset\mathbb{R}^n\times\mathbb{R},
$$

where the structure function b is differentiable with respect to x and satisfies standard ellipticity and growth properties with polynomial growth rate p ∊ (2n/(n + 2), 2). We investigate regularity properties of the solution, including the existence of second-order spatial derivatives, the existence of the time derivative and the higher integrability of the spatial gradient. As an application, we derive dimension estimates for the singular set of solutions of homogeneous parabolic systems. More precisely, we establish the bound

$$\mathrm{dim}_{\mathrm{par}}(\mathrm{sing}\,u)\le n+2-2\beta,$$

provided the structure function depends Höolder continuously on the space variable with Höolder exponent β ∊(0, 1].

(Received October 30 2009)

(Accepted March 01 2010)