MAXIMAL REGULARITY FOR EVOLUTION EQUATIONS GOVERNED BY NON-AUTONOMOUS FORMS

We consider a non-autonomous evolutionary problem (u) over dot(t) + A(t)u(t) = f(t), (u) over dot(0) = u(0) where the operator A(t): V -> V' is associated with a form a(t,.,.) : V x V -> R and u(0) is an element of V. Our main concern is to prove well-posedness with maximal regularity, which means the following. Given a Hilbert space H such that V is continuously and densely embedded into H and given f is an element of L-2(0, T; H), we are interested in solutions u is an element of H-1 (0, T; H) boolean AND L-2(0, T; V). We do prove well-posedness in this sense whenever the form is piecewise Lipschitz-continuous and satisfies the square root property. Moreover, we show that each solution is in C([0, T]; V). The results are applied to non-autonomous Robin-boundary conditions and maximal regularity is used to solve a quasifinear problem.