English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We present a new proof for the equivalence of potential theoretic weaksolutions and viscosity solutions to the $\texttt{p}(\cdot)$-Laplace equationin $\mathbb{R}^n$. The proof of the equivalence in the variable exponent casein Euclidean space was first given by Juutinen, Lukkari, and Parviainen (2010)and extended the equivalence of potential theoretic weak solutions andviscosity solutions to the $\texttt{p}$-Laplace equation in $\mathbb{R}^n$,given by Juutinen, Lindqvist, and Manfredi (2001). In both the fixed exponentcase and the variable exponent case, the main argument is based on the maximumprinciple for semicontinuous functions, several approximations, and alsoapplied the full uniqueness machinery of the theory of viscosity solutions.This paper extends the approach of Julin and Juutinen (2012) for the fixedexponent case, and so we employ infimal convolutions to present a direct, newproof for the equivalence of potential theoretic weak solutions and viscositysolutions to the $\texttt{p}(\cdot)$-Laplace equation in $\mathbb{R}^n$.

A New Proof of the Equivalence of Weak and Viscosity Solutions to the Homogeneous $\texttt{p}(\cdot)$-Laplacian in $\mathbb{R}^n$