Geodesic
$C^1$-extension of subharmonic functions from closed Jordan domains in $\mathbb R^2$
Izvestiya. Mathematics, Tome 68 (2004) no. 6, pp. 1165-1178 
M. S. Mel'nikov; P. V. Paramonov. $C^1$-extension of subharmonic functions from closed Jordan domains in $\mathbb R^2$. Izvestiya. Mathematics, Tome 68 (2004) no. 6, pp. 1165-1178. http://geodesic.mathdoc.fr/item/IM2_2004_68_6_a5/
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     title = {$C^1$-extension of subharmonic functions from closed {Jordan} domains in~$\mathbb R^2$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

For Jordan domains $D$ in $\mathbb R^2$ of Dini–Lyapunov type, we show that any function subharmonic in $D$ and of class $C^1(\overline D)$ can be extended to a function subharmonic and of class $C^1$ on the whole of $\mathbb R^2$ with a uniform estimate of its gradient. We construct a large class of Jordan domains (including domains with $C^1$-smooth boundaries) for which this extension property fails. We also prove a localization theorem on $C^1$-subharmonic extension from any closed Jordan domain.

[1] Verdera Dzh., Melnikov M. S., Paramonov P. V., “$C^1$-approksimatsiya i prodolzhenie subgarmonicheskikh funktsii”, Matem. sb., 192:4 (2001), 37–58 | MR | Zbl

[2] Gardiner S. J., Gustafsson A., “Smooth potentials with prescribed boundary behaviour”, Publ. Matem., 48:1 (2004), 241–249 | MR | Zbl

[3] Vitushkin A. G., “Analiticheskaya emkost mnozhestv v zadachakh teorii priblizhenii”, UMN, 22:6 (1967), 141–199 | MR

[4] Pommerenke Ch., Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992 | MR | Zbl

[5] Stein I., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[6] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1971 | MR | Zbl

[7] Mattila P., Paramonov P. V., “On geometric properties of harmonic $\operatorname{Lip}_1$-capacity”, Pacific J. Math., 171:2 (1995), 469–490 | MR