Non-autonomous maximal regularity for forms of bounded variation

We consider a non-autonomous evolutionary problemu′(t)+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 A(t):V→V′ is associated with a coercive, bounded, symmetric form a(t,.,.):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 whether 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:[0,T]→R such that|a(t,v,w)−a(s,v,w)|≤[g(t)−g(s)]‖v‖V‖w‖V(0≤s≤t≤T,v,w∈V). In that case, we also show that u(.) is continuous with values in V. Moreover we extend this result to certain perturbations of A(t).