Maximal regularity for generalized boundary conditions in time

We consider autonomous and non-autonomous evolution equations on a time interval [0,τ ] [ 0 , τ ] in a Banach space X with the nonstandard time–boundary condition u(0)=Φ u(τ ) u ( 0 ) = Φ u ( τ ) , where Φ Φ is a linear map on X . If Φ =0 Φ = 0 , this is an initial value problem, whereas Φ =I Φ = I corresponds to periodic boundary conditions, and Φ =-I Φ = - I to antiperiodic boundary conditions. Our main point is to establish maximal Lᵖ 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 $$X=H$$ 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 nonstandard time–boundary condition.