Second-Order Regularity for Parabolic p-Laplace Problems

Abstract

Optimal second-order regularity in the space variables is established for solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.Here, p > 1, Ω is an open set in R n , n ≥ 1, with finite Lebesgue measure |Ω|, and T > 0. Moreover, Ω T = Ω × (0, T ), the functions f : Ω T → R N and ψ : Ω → R N , N ≥ 1 are given, and u : Ω → R N is the unknown. According to usage, ∇u stands for the gradient of u with respect to the space variables x ∈ Ω, and u t for its derivative in time t ∈ (0, T ).Mathematics Subject Classifications: 35K20, 35K65, 35B65.

On the second-order regularity of solutions to the parabolic p-Laplace equation

“…As a consequence, he also observed that the time derivative u t exists and belongs to a Sobolev space. See also a recent paper by Cianchi and Maz'ya [5].…”

Global regularity for systems with p-structure depending on the symmetric gradient

“…We refer to the classical results by DiBenedetto [20], Gilbarg and Trudinger [26], Ladyžhenskaja et al [28,29], Liebermann [30], Uhlenbeck [40], Ural'ceva [41], just to cite a few; or the ones linked more to applications Bensoussan and Frehse [9], Nečas [34], and Fuchs and Seregin [24]. Even if the studies started in the sixties, we observe that the field is still extremely active and very recent results are those in [3,4,17,18].…”

Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

“…It is shown that u t ∈ L 2 (Q T ) and |∇u| p−2 ∇u ∈ W 1,2 (Q T ), provided that f ∈ L 2 (Q T ) and u 0 ∈ L 2 (Q T ) ∩ W 1,p 0 (Ω). This regularity result is sharp -see [22,Remark 2.3]. It is worth noting here that the use of Galerkin's approximations prevents one from employing the techniques developed in these works.…”

“…The stronger result on the differentiablity of the flux is obtained in [22]. It is shown that u t ∈ L 2 (Q T ) and |∇u| p−2 ∇u ∈ W 1,2 (Q T ), provided that f ∈ L 2 (Q T ) and u 0 ∈ L 2 (Q T ) ∩ W 1,p 0 (Ω).…”