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

2020/05/04 by Yuzhou Fang, Chao Zhang, Fang, Yuzhou +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.2005.01248

Abstract

We investigate the different notions of solutions to the double-phase equation -\dive(|Du|^p-2Du+a(x)|Du|^q-2Du)=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 A_H(⋅)-harmonic functions of nonlinear potential theory, and then show that A_H(⋅)-harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.

Related

• Equivalence of weak and viscosity solutions for the nonhomogeneous double phase equation
• On weak and viscosity solutions of nonlocal double phase equations
• Asymptotic mean value properties for the elliptic and parabolic double phase equations
• On equivalence of weak and viscosity solutions to nonlocal double phase problems with nonhomogeneous data
• Comparison principles for viscosity solutions of elliptic branches of fully nonlinear equations independent of the gradient