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Singular Domains

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Divergence Operator and Related Inequalities

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Abstract

In view of the previous chapters it is a natural question whether similar results can be obtained for more general domains. With this goal, we consider generalizations of Korn p and div p using weighted Sobolev norms.

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Notes

  1. 1.

    Weak-type is weaker than continuity as an operator \(T: L^{1}(\varOmega ) \rightarrow L^{1}(\varOmega )\) (usually called strong-type (1, 1) in Harmonic Analysis [84]). Indeed, for any λ > 0, the inequality

    $$\displaystyle{\lambda \vert \{x: \vert Tf(x)\vert >\lambda \} \vert \leq \| Tf\|_{L^{1}(\varOmega )},}$$

    is immediate.

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  1. Department of Mathematics and IMAS, University of Buenos Aires and CONICET, Buenos Aires, Argentina

    Gabriel Acosta & Ricardo G. Durán

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Acosta, G., Durán, R.G. (2017). Singular Domains. In: Divergence Operator and Related Inequalities. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6985-2_4

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