Kato's inequality for the strong p(·)-Laplacian

Abstract

Let d∈N and Ω⊂Rd be open. Consider the strong p(⋅)-Laplacian Δ˜p(⋅)u:=|∇u|p(⋅)−4[(p(⋅)−2)Δ∞u+|∇u|2Δu], where Δ∞u:=∑i,j=1d(∂iu)(∂ju)∂2iju. We show that Δ˜p(⋅)|u|≥(sgnu)Δ˜p(⋅)u in the sense of distributions for a certain exponent p∈C1(Ω) with 1<p−<p+<∞ 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(⋅)-Laplacian.