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

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.