2014/06/11 by Dier, Dominik · 1 citation
#35K45 #35K50 #35K90 #47D06 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1406.2884
We consider a non-autonomous evolutionary problem u' (t)+\mathcal A (t)u(t)=f(t), u(0)=u0, where V, H are Hilbert spaces such that V is continuously and densely embedded in H and the operator \mathcal A (t)\colon V→ V^′ is associated with a coercive, bounded, symmetric form \mathfraka(t,.,.)\colon V× V → ℂ for all t ∈ [0,T]. Given f ∈ L2(0,T;H), u0∈ V there exists always a unique solution u ∈ MR(V,V'):= L2(0,T;V) ∩ H1(0,T;V'). The purpose of this article is to investigate when u ∈ H1(0,T;H). This property of maximal regularity in H is not known in general. We give a positive answer if the form is of bounded variation; i.e., if there exists a bounded and non-decreasing function g \colon [0,T] → ℝ such that |\mathfraka(t,u,v)- \mathfraka(s,u,v)| ≤ [g(t)-g(s)] ‖ u ‖V ‖ v ‖V (s,t ∈ [0,T], s ≤ t). In that case, we also show that u(.) is continuous with values in V. Moreover we extend this result to certain perturbations of \mathcal A (t).