Non-autonomous maximal regularity for forms of bounded variation

作者：Dier Dominik^*

来源：Journal of Mathematical Analysis and Applications, 2015, 425(1): 33-54.

DOI：10.1016/j.jmaa.2014.12.006

• Sesquilinear forms
• Non-autonomous evolution equations
• Maximal regularity

We consider a non-autonomous evolutionary problem
u' (t) +A(t)u(t) = f (t), u(0) = u(0),
where V, H are Hilbert spaces such that V is continuously and densely embedded in H and the operator A(t): V -> V' is associated with a coercive, bounded, symmetric form a(t,.,.): V x V -> C for all t epsilon [0,T]. Given f epsilon L-2 (0, T; H), u(0) epsilon V there exists always a unique solution u epsilon MR(V, V') := L-2(0, T;V) boolean AND H-1(0,T;V'). The purpose of this article is to investigate whether u epsilon H-1(0, T; H). This property of maximal regularity in H is not known in general. We give a positive answer if the form is of bounded variation; i.e., if there exists a bounded and non-decreasing function g: [0, T] -> R such that
vertical bar a(t, upsilon, w) a(s, upsilon, w)vertical bar <= [g(t) - g((S)]parallel to upsilon parallel to V parallel to w parallel to V (0 <= s <= t <= T, upsilon, w epsilon V
In that case, we also show that u(.) is continuous with values in V. Moreover we extend this result to certain perturbations of A(t).

• 出版日期2015-5-1