A+ CATEGORY SCIENTIFIC UNIT

Kato’s inequality for the strong $p(\cdot)$-Laplacian

Tan Duc Do, Le Xuan Truong Studia Mathematica MSC: Primary 35J70; Secondary 35J60, 35B20. DOI: 10.4064/sm220330-19-9 Published online: 17 October 2022

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.

Authors

  • Tan Duc DoUniversity of Economics Ho Chi Minh City (UEH)
    Ho Chi Minh City, Vietnam
    e-mail
  • Le Xuan TruongUniversity of Economics Ho Chi Minh City (UEH)
    Ho Chi Minh City, Vietnam
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image