We study a mixed boundary value problem for the $p$-Laplace equation ${\Delta }_{p}u=0$ in an open infinite circular half-cylinder with prescribed Dirichlet boundary data on a part of the boundary and zero Neumann boundary data on the rest. Existence of weak solutions to the mixed problem is proved both for Sobolev and for continuous data on the Dirichlet part of the boundary. We also obtain a boundary regularity result for the point at infinity in terms of a variational capacity adapted to the cylinder.

中文翻译：

---

的混合边值问题 $p$无限圆柱体中的谐波函数

我们研究了混合边值问题 $p$-拉普拉斯方程 ${\Delta }_pü=0$在一个开放的无限圆形半圆柱体中，在一部分边界上具有规定的Dirichlet边界数据，而在其余边界上具有零Neumann边界数据。Sobolev和边界Dirichlet部分上的连续数据都证明了混合问题的弱解的存在。我们还根据适合圆柱体的变化能力，获得了无穷大点的边界规则性结果。