Maximal regularity for generalized boundary conditions in time

TL;DR

This paper establishes maximal $L^p$-regularity for evolution equations with non-standard boundary conditions involving a linear map, covering autonomous and non-autonomous cases, including periodic and anti-periodic conditions.

Contribution

It extends maximal regularity results to boundary conditions defined by a linear map, including non-autonomous operators and forms, in both Banach and Hilbert spaces.

Findings

Maximal $L^p$-regularity is proven for these generalized boundary conditions.

Results cover autonomous and non-autonomous evolution equations.

Special focus on regularity in Hilbert spaces with non-standard boundary conditions.

Abstract

We consider autonomous and non-autonomous evolution equations on a time interval $[0,\tau]$ in a Banach space $X$ with the non-standard time-boundary condition $u(0)=\Phi u(\tau)$, where $\Phi$ is a linear map on $X$. If $\Phi=0$, this is an initial value problem, whereas $\Phi=I$ corresponds to periodic boundary conditions, and $\Phi=-I$ 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.