Sufficient Conditions for Integrability of Distortion Function Kf 1

Capozzoli, Costantino

Bollettino dell'Unione Matematica Italiana, Tome 2 (2009), p. 699-710 / Harvested from Biblioteca Digitale Italiana di Matematica

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Résumé

Assume that $\Omega$ , $\Omega^{\prime}$ are planar domains and $f\colon\Omega\xrightarrow{\text{onto}}\Omega^{\prime}$ is a homeomorphism belonging to Sobolev space $W_{\text{loc}}^{1,1}(\Omega;\mathbb{R}^{2})$ with finite distortion. We prove that if the distortion function $K_{f}$ of $f$ satisfies the condition $\operatorname{dist}_{\text{EXP}}(K_{f},L^{\infty})<1$ , then the distortion function $K_{f^{-1}}$ of $f^{-1}$ belongs to $L^{1}_{\text{loc}}(\Omega^{\prime})$ . We show that this result is sharp in sense that the conclusion fails if $\operatorname{dist}_{\text{EXP}}(K_{f},L^{\infty})=1$ . Moreover, we prove that if the distortion function $K_{f}$ satisfies the condition $\operatorname{dist}_{\text{EXP}}(K_{f},L^{\infty})=\lambda$ for some $\lambda>0$ , then $K_{f^{-1}}$ belongs to $L^{p}_{\text{loc}}(\Omega^{\prime})$ for every $p\in(0,\frac{1}{2\lambda})$ . As special case of this result we show that if the distortion function $K_{f}$ satisfies the condition $\operatorname{dist}_{\text{EXP}}(K_{f},L^{\infty})=0$ , then $K_{f^{-1}}$ belongs to intersection of $L^{p}_{\text{loc}}(\Omega^{\prime})$ for all $p>1$ .

Publié le : 2009-10-01

MR 2569298 | Zbl 1191.46027

@article{BUMI_2009_9_2_3_699_0,
     author = {Costantino Capozzoli},
     title = {Sufficient Conditions for Integrability of Distortion Function Kf 1},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {2},
     year = {2009},
     pages = {699-710},
     zbl = {1191.46027},
     mrnumber = {2569298},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2009_9_2_3_699_0}
}

Capozzoli, Costantino. Sufficient Conditions for Integrability of Distortion Function Kf 1. Bollettino dell'Unione Matematica Italiana, Tome 2 (2009) pp. 699-710. http://gdmltest.u-ga.fr/item/BUMI_2009_9_2_3_699_0/

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