Harmonic functions which vanish on a cylindrical surface
Let \(a>0\), \(N\geq 3\) and \(\Omega_a\) be the finite cylinder \(B'\times (-a,a)\), where \(B'\) is the open unit ball in \({\mathbb R}^{N-1}\). The main result of this paper establishes that any harmonic function \(h\) on \(\Omega_a\) that vanishes continuously on \(\partial B'\times (-a,a)\) has a harmonic extension \(\tilde h\) to \({\mathbb R}^{N-1}\times (-a,a)\). The authors also establish the following estimate: for any \(b\in (0,a)\) there is a constant \(c=c(N,a,b,h)\) such that \(|\tilde h(x)|\leq c\|x'\|^{1-N/2}\) for all \(x=(x',x_N)\in ({\mathbb R}^{N_1}\setminus B')\times (-b,b)\). The approach relies on the analysis of the convergence properties of double series expansion of the Green function for an infinite cylinder.
scientific article; zbMATH DE number 6487342