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.
Melnikov, M. S., & Paramonov, P. V. (2004). C 1-extension of subharmonic functions from closed Jordan domains in ℝ 2. Izvestiya Mathematics, 68(6), 1165-1178. https://doi.org/10.1070/IM2004v068n06ABEH000514