Equivalence between distributional and viscosity solutions for the double-phase equation

TL;DR

This paper demonstrates that for the double-phase equation, the notions of distributional and viscosity solutions are equivalent by introducing a new class of harmonic functions in nonlinear potential theory.

Contribution

The paper establishes the equivalence between distributional and viscosity solutions for the double-phase equation using the concept of _{H(\u2208)}-harmonic functions.

Findings

Distributional and viscosity solutions coincide for the double-phase equation.

Introduction of _{H()}-harmonic functions in nonlinear potential theory.

Equivalence result enhances understanding of solution concepts for variable ellipticity equations.

Abstract

We investigate the different notions of solutions to the double-phase equation $$ -\dive(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=0, $$ which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the $\mathcal{A}_{H(\cdot)}$-harmonic functions of nonlinear potential theory, and then show that $\mathcal{A}_{H(\cdot)}$-harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.