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.
Abstract. We use the theory of boundary values (also called traces) of holomorphic semigroups as developed by Boyadzhiev-deLaubenfels (1993) and El-Mennaoui (1992) to study the second order Cauchy problem for certain generators of holomorphic semigroups. Our results contain in particular the result of Hieber (Math. Ann. 291 (1991), 1-16) for the Laplace operator on L p (R N ).
We prove the validity of the inverse formula of the Laplace transform for C 0 -semigroups in UMD-spaces in the strong sense and give an example which shows that the UMD-property is essential.
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