Extension theorems for harmonic functions which vanish on a subset of a cylindrical surface (Q2668941)

The goal of this article is to obtain extension results for functions which are harmonic on a subdomain of an infinite cylinder and vanish on a relatively open subset of the cylindrical surface. Let \(B'\) denote the unit ball of \(\mathbb{R}^{N-1}\), \(N\geq 3\), and let \(\Omega=B'\times \mathbb{R}\) be the infinite cylinder in \(\mathbb{R}^N\). For a closed set \(E\subset \overline \Omega\), denote \(d_t^E(x')=\inf\big\{ \|x'-y'\|; (y',t)\in E\big\}\), \(x'\in B'\), \(t\in \mathbb{R}\). The main results of the article are as follows. Theorem 1. Let \(\omega\) be a relatively open subset of \(\partial B'\times (-a,a)\), where \(a>0\). Then, any harmonic function on \(B'\times (-a,a)\) which continuously vanishes on \(\omega\) has a harmonic extension to the set \[ \big\{(\rho x',t): x'\in \partial B' , |t|<a, 0\leq \rho<1+d_t^{\partial \Omega\setminus \omega}(x') \big\}. \] Theorem 2. Let \(\omega\) be a relatively open subset of \(\partial B'\times (-a,a)\) and \(\Phi:(-a,a)\to [0,1)\) be continuous. Then, any harmonic function on \(R_\Phi(\omega):=\big\{(\rho x',t): (x',t)\in \omega, \Phi(t)<\rho<1\big\}\) which continuously vanishes on \(\omega\) has a harmonic extension to the set \[ \big\{(\rho x',t): x'\in \partial B' , |t|<a, \Phi(t)< \rho<1+d_t^{\partial \Omega\setminus R_\Phi(\omega)}(x') \big\}. \]