Maximal regularity for generalized boundary conditions in time

Abstract

We consider autonomous and non-autonomous evolution equations on a time interval $[0,\ddot{I}„]$ in a Banach space $X$ with the non-standard time-boundary condition $u(0)=\hat{I}¦u(\ddot{I}„)$, where $\hat{I}¦$ is a linear map on $X$. If $\hat{I}¦=0$, this is an initial value problem, whereas $\hat{I}¦=I$ corresponds to periodic boundary conditions, and $\hat{I}¦=-I$ to antiperiodic boundary conditions. Our main point is to establish maximal $L^p$-regularity. In the non-autonomous case we consider two situations. The first concerns time-dependent operators with a fixed domain. In the second one we take $X=H$ a Hilbert space and consider evolution equations associated with non-autonomous forms. Of special interest is then maximal regularity in $H$ with a non-standard time-boundary condition.

28 pages