On the viscosity solutions to Trudinger’s equation

Abstract

We study the existence of positive viscosity solutions to Trudinger's equation for cylindrical domains Ω × [0, T ), where Ω ⊂ IR n , n ≥ 2, is a bounded domain, T > 0 and 2 ≤ p < ∞. We show existence for general domains Ω, when n < p < ∞. For 2 ≤ p ≤ n, we prove existence for domains Ω that satisfy a uniform outer ball condition. We achieve this by constructing suitable sub-solutions and super-solutions and applying Perron's method.

Asymptotics of viscosity solutions to some doubly nonlinear parabolic equations

“…The main focus of this paper is Trudinger's equation but we will also state some results for a parabolic equation involving the infinity-Laplacian. This is a followup of the works in [4,5].…”

“…In [4] (see Theorem 5.2), we showed the existence of positive viscosity solutions of (1.5) for p = ∞. The work [5] showed the existence of positive viscosity solutions for 2 ≤ p < ∞, see Theorems 1.1 and 1.2 therein. For the case 2 ≤ p ≤ n, this result is proven for domains Ω that satisfy a uniform outer ball condition.…”

“…The equation Γ p = 0, 2 ≤ p < ∞, is the well-known Trudinger equation [11]. See also [4,5] and the references therein. The operators Γ p , 2 < p ≤ ∞ are doubly nonlinear and degenerate and, in this work, solutions will be understood to be in the viscosity sense.…”

On a comparison principle for Trudinger’s equation

“…The equation and its regularity have also been treated in [10,11,[19][20][21]23]. See also [2] for a viscosity approach to the equation. Plan of the paper.…”

On the viscosity solutions to a class of nonlinear degenerate parabolic differential equations

“…A similar quotient type comparison principle was derived for the doubly nonlinear parabolic equations studied in [1,2]. In Sections 5, 6 and 7, we address the existence of positive solutions to (1.1), i.e,…”