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

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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. 
@article{IM2_2004_68_6_a5,
     author = {M. S. Mel'nikov and P. V. Paramonov},
     title = {$C^1$-extension of subharmonic functions from closed {Jordan} domains in~$\mathbb R^2$},
     journal = {Izvestiya. Mathematics },
     pages = {1165--1178},
     publisher = {mathdoc},
     volume = {68},
     number = {6},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2004_68_6_a5/}
}
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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/