Equivalence of weak and viscosity solutions for the nonhomogeneous double phase equation

TL;DR

This paper proves the equivalence between weak and viscosity solutions for a class of nonhomogeneous double phase equations, establishing conditions under which solutions are Lipschitz continuous and demonstrating the use of convolution techniques.

Contribution

It introduces new hypotheses on coefficients and exponents to establish the equivalence and regularity of solutions for the nonhomogeneous double phase equation.

Findings

Viscosity solutions with Lipschitz continuity are weak solutions.

Bounded viscosity solutions are Lipschitz continuous.

Equivalence holds under specific conditions on coefficients and exponents.

Abstract

We establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term $$ -{\rm div}(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=f(x,u,Du),\quad 1<p\le q<\infty, a(x)\ge0. $$ We find some appropriate hypotheses on the coefficient $a(x)$, the exponents $p, q$ and the nonlinear term $f$ to show that the viscosity solutions with {\em a priori} Lipschitz continuity are weak solutions of such equation by virtue of the $\inf$($\sup$)-convolution techniques. The reverse implication can be concluded through comparison principles. Moreover, we verify that the bounded viscosity solutions are exactly Lipschitz continuous, which is also of independent interest.