Maximal L p regularity for parabolic difference equations

Abstract In this paper, we consider a family of finite difference operators { Ah } h >0 on discrete L q ‐spaces L q (ℝ Nh ). We show that the solution u h to u ′ h ( t ) – A h u h ( t ) = f h ( t ), t > 0, u h (0) = 0 satisfies the estimate ‖ A h u h ‖   L  p(ℝ   + ; L  q(ℝ   N  h))≤ C ‖ f h ‖   L  p(ℝ   + ; L  q(ℝ   N  h)) , where C is independent of h and f h . In this case, the family { A h } h >0 is said to have discrete maximal L p regularity on the discrete L q ‐space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)