A criterion for the regularity of a point at infinity for the Zaremba problem in a semicylinder (Q1113007)

Let G be a half-cylinder \(\{x=(x',x_ n):\) \(x_ n>0\); x'\(\in \omega \}\), where \(\omega\) is a domain in \(R^{n-1}\) with smooth boundary and with a compact closure. Let F be a closed subset of \(\partial G\) with a limit point at infinity. The Laremba boundary value problem on G with the Neumann data on \(\partial G\setminus F\) and with the Dirichlet data \(u=0\) on F is considered. A point at infinity is called regular if the solution tends to zero as \(x_ n\to \infty\), \(x\in G\) for any Neumann data with compact support. It is proved that the regularity is equivalent to \[ \sum^{\infty}_{j=1}j cap(\{x\in F:\quad j\leq x_ n\leq j+1\})=\infty, \] where cap is the Wiener capacity. Pointwise estimates for solutions, the Green function and the harmonic measure are obtained in terms of cap.