# Directional ellipticity on special domains: weak Maximum and Phragm\`en-Lindel\"of principles

## TL;DR

This paper establishes maximum principles for a class of fully nonlinear, possibly degenerate elliptic operators on unbounded cylindrical domains, advancing understanding of elliptic PDE behavior in complex geometries.

## Contribution

It proves maximum principles for nonlinear operators on unbounded cylindrical domains with ellipticity only in bounded directions, allowing degeneracy along unbounded directions.

## Key findings

- Maximum principles hold under specified structural conditions.
- Results apply to fully nonlinear operators with degeneracy.
- Advances understanding of PDEs on unbounded domains.

## Abstract

We prove the validity of maximum principles for a class of fully nonlinear operators on unbounded subdomains $\Omega \subset \mathbb R^n$ of cylindrical type. The main structural assumption is the uniform ellipticity of the operator along the bounded directions of $\Omega$, with possible degeneracy along the unbounded directions.