On higher differentiability of solutions of parabolic systems with discontinuous coefficients and (p,q)-growth

Abstract

AbstractWe consider weak solutions$u:\Omega _T\to {\open R}^N$to parabolic systems of the type$$u_t-{\rm div}\;a(x,t,Du) = 0\quad {\rm in}\;\Omega _T = \Omega \times (0,T),$$where the functiona(x,t,ξ) satisfies (p,q)-growth conditions. We give an a priori estimate for weak solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps$x\mapsto a(x,t,\xi )$under consideration may not be continuous, but may only possess a Sobolev-type regularity. In a certain sense, our assumption means that the weak derivatives$D_xa(\cdot ,\cdot ,\xi )$are contained in the class$L^\alpha (0,T;L^\beta (\Omega ))$, where the integrability exponents$\alpha ,\beta $are coupled by$$\displaystyle{{p(n + 2)-2n} \over {2\alpha }} + \displaystyle{n \over \beta } = 1-\kappa $$for someκ∈ (0,1). For the gap between the two growth exponents we assume$$2 \les p < q \les p + \displaystyle{{2\kappa } \over {n + 2}}.$$Under further assumptions on the integrability of the spatial gradient, we prove a result on higher differentiability in space as well as the existence of a weak time derivative$u_t\in L^{p/(q-1)}_{{\rm loc}}(\Omega _T)$. We use the corresponding a priori estimate to deduce the existence of solutions of Cauchy–Dirichlet problems with the mentioned higher differentiability property.