Kato’s inequality for the strong $p(\cdot )$-Laplacian
Volume 270 / 2023
Abstract
Let $d \in \mathbb N$ and $\Omega \subset \mathbb R^d$ be open. Consider the strong $p(\cdot )$-Laplacian \[ \widetilde{\Delta }_{p(\cdot )} u := |\nabla u|^{p(\cdot )-4} [ ( p(\cdot ) - 2 ) \Delta _\infty u + |\nabla u|^2 \Delta u ], \] where \[ \Delta _\infty u := \sum _{i,j=1}^d (\partial_i u) (\partial_j u) \partial_{ij}^2 u. \] We show that \[ \widetilde {\Delta }_{p(\cdot )} |u| \ge ({\rm sgn}\, u) \widetilde {\Delta }_{p(\cdot )} u \] in the sense of distributions for a certain exponent $p \in C^1(\Omega )$ with $1 \lt p^- \lt p^+ \lt \infty $ and for functions $u$ belonging to an admissible class. This extends the well-known Kato’s inequality for strongly elliptic second-order differential operators to the strong $p(\cdot )$-Laplacian.
