On the viscosity solutions to Trudinger's equation

TL;DR

This paper investigates the existence of positive viscosity solutions to Trudinger's equation in cylindrical domains, establishing existence results under various conditions on the domain and the parameter p.

Contribution

The paper provides new existence results for viscosity solutions to Trudinger's equation in bounded domains, using Perron's method and constructing sub- and super-solutions.

Findings

Existence for general domains when n < p < ∞.

Existence for domains with uniform outer ball condition when 2 ≤ p ≤ n.

Construction of sub- and super-solutions for applying Perron's method.

Abstract

We study the existence of positive viscosity solutions to Trudinger's equation for cylindrical domains $\Omega\times[0, T)$, where $\Omega\subset \mathbb{R}^n,\;n\ge 2,$ is a bounded domain, $T>0$ and $2\leq p<\infty$. We show existence for general domains $\Omega,$ when $n<p<\infty$. For $2\leq p\leq n$, we prove existence for domains $\Omega$ that satisfy a uniform outer ball condition. We achieve this by constructing suitable sub-solutions and super-solutions and applying Perron's method.