Maximal regularity for non-autonomous Robin boundary conditions

Abstract

We consider a non-autonomous Cauchy problem

where $A\left(t\right)$ is associated with the form $a\left(t;.,.\right):VxV\to ℂ$ , where $V$ and $H$ are Hilbert spaces such that $V$ is continuously and densely embedded in $H$. We prove $H$-maximal regularity, i.e., the weak solution $u$ is actually in $H^1\left(0,T;H\right)$ (if $u_0\in V$ and $f\in L^2\left(0,T;H\right)$) under a new regularity condition on the form $a$ with respect to time; namely Hölder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.