Results on maximal regularity in the non-autonomous case

Sylvie Monniaux Mathematik V, Universität Ulm 89069 Ulm Germany

             monniaux@mathematik.uni-ulm.de

It is known that the L^{p}-maximal regularity for a generator of an holomorphic semigroup on a Banach space is independent of p. In the case of an Hilbert space, there is always maximal regularity for this class of operators. Moreover, Hieber/Prüss and Coulhon/Duong showed that if the semigroup verifies a certain heat kernel estimate on L^{2}, then we also have L^{p}-maximal regularity on all L^{q}. We show that the same results are true if we consider the non-autonomous Cauchy problem

u′(t) + A(t)u(t) = f(t), u(0) = 0

for a family A(t), t ∈ [0, T], where the operators −A(t) generate holomorphic semigroups on a Banach space and verify a certain regularity condition.