Equivalence of solutions for non-homogeneous p(x)-Laplace equations

TL;DR

This paper proves the equivalence between weak and viscosity solutions for non-homogeneous p(x)-Laplace equations, addressing challenges posed by variable exponents, log-terms, and non-invariance under translations.

Contribution

It establishes the equivalence between weak and viscosity solutions for p(x)-Laplace equations with variable coefficients, using inf- and sup-convolution techniques and comparison principles.

Findings

Viscosity solutions are also weak solutions.

Weak solutions are viscosity solutions under certain conditions.

Addresses challenges due to variable exponents and non-invariance.

Abstract

We establish the equivalence between weak and viscosity solutions for non-homogeneous $p(x)$-Laplace equations with a right-hand side term depending on the spatial variable, the unknown, and its gradient. We employ inf- and sup-convolution techniques to state that viscosity solutions are also weak solutions, and comparison principles to prove the converse. The new aspects of the $p(x)$-Laplacian compared to the constant case are the presence of $\log$-terms and the lack of the invariance under translations.