We consider autonomous and non-autonomous evolution equations on a time interval <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$[0,\tau ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> in a Banach space X with the nonstandard time–boundary condition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u(0)=\Phi u(\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> is a linear map on X. If <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, this is an initial value problem, whereas <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Phi =I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> corresponds to periodic boundary conditions, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Phi =-I$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mo>-</mo> <mi>I</mi> </mrow> </math></EquationSource> </InlineEquation> to antiperiodic boundary conditions. Our main point is to establish maximal <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-regularity. In the non-autonomous case we consider two situations. The first concerns time-dependent operators with a fixed domain. In the second one we take <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$X=H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> 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.

Maximal regularity for generalized boundary conditions in time

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2025年11月15日

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Wolfgang Arendt,
Manfred Sauter

摘要

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