Mixed boundary value problem for $p$-harmonic functions in an infinite cylinder

TL;DR

This paper investigates the existence and boundary regularity of solutions to a mixed boundary value problem for the p-Laplace equation in an infinite cylindrical domain, extending understanding of nonlinear PDEs in unbounded geometries.

Contribution

It establishes the existence of weak solutions for both Sobolev and continuous data and provides a boundary regularity result at infinity using a specialized variational capacity.

Findings

Existence of weak solutions for mixed boundary conditions.

Boundary regularity at infinity characterized by a variational capacity.

Results applicable to both Sobolev and continuous boundary data.

Abstract

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.