2016/03/02 by Omar, EL-Mennaoui, Hafida, Laasri
#35K45 #35K50 #35K90 #47D06 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1603.01100
We consider a linear non-autonomous evolutionary Cauchy problem u (t)+A(t)u(t)=f(t) \hbox for \hboxa.e. t∈ [0,T], u(0)=u0, where the operator A(t) arises from a time depending sesquilinear form a(t,.,.) on a Hilbert space H with constant domain V. Recently a result on L2-maximal regularity in H, i.e., for each given f∈ L2(0,T,H) and u0 ∈ V the problem above has a unique solution u∈ L2(0,T,V)∩ H1(0,T,H), is proved in [10] under the assumption that a is symmetric and of bounded variation. The aim of this paper is to prove that the solutions of an approximate non-autonomous Cauchy problem in which a is symmetric and piecewise affine are closed to the solutions of that governed by symmetric and of bounded variation form. In particular, this provide an alternative proof of the result in [10] on L2-maximal regularity in H.