Harmonic extension through conical surfaces (Q2089721)

The paper under review deals with extension results for harmonic functions vanishing on a conical surface and are based on expansions for the Green function of an infinite cone. More precisely, they show that if \(0<\theta_\ast<\pi\) and if \(h\) is an harmonic function on the infinite cone \[ \Omega(\theta_\ast)\equiv \{x=(x',x_N)\in \mathbb{R}^{N-1}\times \mathbb{R}\colon \cos^{-1}(x_N/\|x\|)<\theta_\ast \} \] vanishing on the boundary \(\partial \Omega(\theta_\ast)\) of \(\Omega(\theta_\ast)\), then \(h\) admits an extension to the set \[ \Omega(\pi)\equiv \{x=(x',x_N)\in \mathbb{R}^{N-1}\times \mathbb{R}\colon \cos^{-1}(x_N/\|x\|)<\pi \}=\mathbb{R}^N \setminus (\{0'\}\times (-\infty,0])\, . \]