Geodesic
$C^m$-extension of subharmonic functions
Izvestiya. Mathematics , Tome 69 (2005) no. 6, pp. 1211-1223.

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Given $m\in(1,3)$ and any (Jordan) $B$-domain $D$ in $\mathbb R^2$, we prove that any function of class $C^m(\,\overline D\,)$ that is subharmonic in $D$ can be extended to a function of class $C^m$ that is subharmonic on the whole $\mathbb R^2$ and give an estimate of the $C^{m-1}$-norm of its gradient. The corresponding assertion for $m\in[0,1)\cup[3,+\infty)$ is false even for discs. These results also hold for balls $D$ in $\mathbb R^N$, $N\in\{3,4,\dots\}$. We also obtain some corollaries, including the corresponding assertions on the $\operatorname{Lip}^m$-extension of subharmonic functions. 
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     title = {$C^m$-extension of subharmonic functions},
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P. V. Paramonov. $C^m$-extension of subharmonic functions. Izvestiya. Mathematics , Tome 69 (2005) no. 6, pp. 1211-1223. http://geodesic.mathdoc.fr/item/IM2_2005_69_6_a7/

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