Higher differentiability of solutions of parabolic systems with discontinuous coefficients
Flavia Giannetti, Antonia Passarelli di Napoli, Christoph Scheven · Journal of the London Mathematical Society · 2016
We consider weak solutions u : Ω T → ℝ N to parabolic systems of the type u t − div a ( x , t , D u ) = 0 in Ω T = Ω × ( 0 , T ) , where the function a ( x , t , ξ ) satisfies standard p-growth and ellipticity conditions for p ⩾ 2 with respect to the gradient variable ξ. We study the regularity of the solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps x ↦ a ( x , t , ξ ) 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 x a ( ⋅ , ⋅ , ξ ) are contained in the class L α ( 0 , T ; L β ( Ω ) ) , where the integrability exponents α , β are coupled by p ( n + 2 ) − 2 n 2 α + n β = 1. In the particular case α = β = p ( n + 2 ) / 2 , our assumption reduces to D x a ∈ L loc p ( n + 2 ) / 2 ( Ω T ) . The aim of this paper is to prove a higher differentiability result of the solutions in the spatial directions as well as the existence of a weak time derivative u t ∈ L loc p / ( p − 1 ) ( Ω T ) .