Maximal regularity for generalized boundary conditions in time

Abstract We consider autonomous and non-autonomous evolution equations on a time interval $$[0,\tau ]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> in a Banach space X with the nonstandard time–boundary condition $$u(0)=\Phi u(\tau )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>Φ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>τ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$\Phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Φ</mml:mi> </mml:math> is a linear map on X . If $$\Phi =0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , this is an initial value problem, whereas $$\Phi =I$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:math> corresponds to periodic boundary conditions, and $$\Phi =-I$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:math> to antiperiodic boundary conditions. Our main point is to establish maximal $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -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 $$X=H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> 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.