$C^1$-extension of subharmonic functions from closed Jordan domains in $\mathbb R^2$

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.