On evolution equations governed by non-autonomous forms

Abstract

We consider a linear non-autonomous evolutionary Cauchy probleṁ1) 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 L 2 -maximal regularity in H, i.e. for each given f ∈ L 2 (0, T, H) and u0 ∈ V the problem (0.1) has a unique solution u ∈ L 2 (0, T, V ) ∩ H 1 (0, T, H), is proved in Dier (J. Math. Anal. Appl. 425:33-54, 2015) 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 nonautonomous 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 provides an alternative proof of the result in Dier (J. Math. Anal. Appl. 425:33-54, 2015) on L 2 -maximal regularity in H. Mathematics Subject Classification. 35K90, 35K50, 35K45, 47D06.

Non-autonomous maximal regularity for forms given by elliptic operators of bounded variation

“…An easier example of the same index is however partially understood. Namely, maximal L 2 -regularity was shown by Dier assuming W 1,1 -regularity and symmetry of the forms [Die15] (for an alternative proof see [EL16]). In fact, the result even holds for forms of bounded variation.…”

A note on the norm-continuity for evolution families arising from non-autonomous forms

“…. This result has been proved by Arendt and Monniaux [4] (see also [10]) when the form a satisfies the weaker condition…”

Regularity properties for evolution families governed by non-autonomous forms

“…Lemma 2.2. The operator P s is well defined and P s L(C(s,T,V )) < 1 4 by choosing c 0 > 0 in (22) to be small enough.…”

“…620] where a Galerkin's method is used or in [39,Section 5.5] where a fundamental solution has been constructed. Furthermore, the author gave an alternative proof in [34,Section 2] using the approach of the integral product of semigroups developed in [20,21,22]. Lions' theorem requires only measurability of t → a(t, u, v) for all u, v ∈ V .…”

“…However, under additional regularity assumptions on the form a, the initial value u 0 and the inhomogeneity f, some positive results were already done by Lions [13]. More recently, this problem has been studied with some progress and different approaches [9,10,29,17,31,32,22,18,12,24]. Results on multiplicative perturbation are established in [9,17,11].…”