For Jordan domains D in ℝ 2 of Dini-Lyapunov type, we show that any function subharmonic in D and of class C 1 (D̄) can be extended to a function subharmonic and of class C 1 on the whole of 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. © 2004 RAS(DoM) and LMS.
|Publication status||Published - 1 Nov 2004|