Behaviour at infinity for solutions of a mixed boundary value problem via inversion

TL;DR

This paper investigates the behavior of solutions to a mixed boundary value problem for a quasilinear elliptic equation in an infinite cylindrical domain, focusing on existence, uniqueness, and asymptotic behavior at infinity.

Contribution

It establishes existence and uniqueness of bounded solutions and characterizes their behavior at infinity using capacity, extending classical principles to this setting.

Findings

Solutions with Neumann data at infinity exhibit three distinct asymptotic behaviors.

The regularity of the point at infinity is characterized via apacitance.

The problem extends classical boundary value analysis to unbounded cylindrical domains.

Abstract

We study a mixed boundary value problem for the quasilinear elliptic equation $\mathop{\rm div}\mathcal{A}(x,\nabla u(x))=0$ in an open infinite circular half-cylinder with prescribed continuous Dirichlet data on a part of the boundary and zero conormal derivative on the rest. We prove the existence and uniqueness of bounded weak solutions to the mixed problem and characterize the regularity of the point at infinity in terms of \p-capacities. For solutions with only Neumann data near the point at infinity we show that they behave in exactly one of three possible ways, similar to the alternatives in the Phragm\'en--Lindel\"of principle.