Maximal L p -regularity for evolution equations governed by non-autonomous forms in weighted spaces

Abstract

We consider the maximal regularity for non-autonomous Cauchy problem u ′ ( t ) + A ( t ) u ( t ) = f ( t ) , t -a.e. , u ( 0 ) = u 0 . Each operator A ( t ) arises from a time depending on a sesquilinear form a ( t ) on a Hilbert space H with a constant domain V . We show maximal L p − regularity result in temporally weighted L p -spaces for 2 < p < ∞ and other regularity properties for the solution of the previous problem under sufficient regularity assumption on the forms and the initial value u 0 , which is significantly weaker than those from previous contributions. Our results are motivated by boundary value problems.