A picone Identity for variable exponent operators and applications

TL;DR

This paper introduces a novel Picone identity for anisotropic quasilinear operators with variable exponents, leading to new inequalities, uniqueness results, and comparison principles for related elliptic and parabolic equations.

Contribution

It extends the Picone identity to variable exponent operators, enabling new analytical tools for quasilinear elliptic and parabolic equations.

Findings

Established a new Picone identity for $p(x)$-Laplacian.

Derived a Diaz-Saa type inequality for variable exponents.

Proved a weak comparison principle for certain parabolic equations.

Abstract

In this work, we establish a new Picone identity for anisotropic quasilinear operators, such as the $p(x)$-Laplacian defined as $\mbox{div}(|\nabla u|^{p(x)-2} \nabla u).$ Our extension provides a new version of the Diaz-Saa inequality and new uniqueness results to some quasilinear elliptic equations with variable exponents. This new Picone identity can be also used to prove some accretivity property to a class of fast diffusion equations involving variable exponents. Using this, we prove for this class of parabolic equations a new weak comparison principle.