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<p>We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Omega \subseteq \mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> that are uniformly <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(C^{2,1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. Starting with the Hodge Laplacian we etablish a bounded Hörmander functional calculus for the Stokes operator with Hodge boundary conditions. This entails a Hörmander functional calculus and boundedness of the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(H^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-calculus in spaces of soleniodal vector fields for the Stokes operator with Hodge boundary conditions. We then establish boundedness of the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(H^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-calculus for Stokes operators with Navier type conditions via Robin type perturbations of Hodge boundary conditions. This implies maximal <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-regularity for these operators and results on fractional domain spaces. Our results cover certain non-Helmholtz domains.</p>

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\(H^\infty \)-calculus for the Stokes operator with Hodge, Navier, and Robin boundary conditions on unbounded domains

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  • 发表日期：

    2025年9月26日

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  • Peer Christian Kunstmann

摘要

We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains \(\Omega \subseteq \mathbb {R}^d\) Ω ⊆ R d that are uniformly \(C^{2,1}\) C 2 , 1 . Starting with the Hodge Laplacian we etablish a bounded Hörmander functional calculus for the Stokes operator with Hodge boundary conditions. This entails a Hörmander functional calculus and boundedness of the \(H^\infty \) H ∞ -calculus in spaces of soleniodal vector fields for the Stokes operator with Hodge boundary conditions. We then establish boundedness of the \(H^\infty \) H ∞ -calculus for Stokes operators with Navier type conditions via Robin type perturbations of Hodge boundary conditions. This implies maximal \(L^p\) L p -regularity for these operators and results on fractional domain spaces. Our results cover certain non-Helmholtz domains.