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Sharp Trace and Korn Inequalities for Differential Operators

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2024

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Diening, Lars

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Potential Analysis. Springer. ISSN 0926-2601. eISSN 1572-929X. Verfügbar unter: doi: 10.1007/s11118-024-10165-1

Zusammenfassung

We establish sharp trace and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov boundary traces by Riesz potentials and oscillations that equally applies to p = 1, a case difficult to be handled by harmonic analysis techniques, we then classify boundary trace- and Korn-type inequalities. For p = 1 and so despite the failure of the Calderón-Zygmund theory, we prove that sharp trace estimates can be systematically reduced to full k-th order gradient estimates. Moreover, for 1 < p < ∞, where sharp trace estimates yield Korn-type inequalities on smooth domains, we show for the basically optimal class of John domains that Korn-type inequalities persist – even though the reduction to global Calderón-Zygmund estimates by extension operators might not be possible.

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Fachgebiet (DDC)
510 Mathematik

Schlagwörter

Trace theorem, Uspenskiĭ’s theorem, Korn inequality, Riesz potentials, Inverse estimates, NTA domains, John domains, Weighted inequalities

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ISO 690DIENING, Lars, Franz GMEINEDER, 2024. Sharp Trace and Korn Inequalities for Differential Operators. In: Potential Analysis. Springer. ISSN 0926-2601. eISSN 1572-929X. Verfügbar unter: doi: 10.1007/s11118-024-10165-1
BibTex
@article{Diening2024-10-02Sharp-71206,
  year={2024},
  doi={10.1007/s11118-024-10165-1},
  title={Sharp Trace and Korn Inequalities for Differential Operators},
  issn={0926-2601},
  journal={Potential Analysis},
  author={Diening, Lars and Gmeineder, Franz}
}
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