The boundary Harnack inequality for variable exponent $p$-Laplacian, Carleson estimates, barrier functions and $p(\cdot)$-harmonic measures

TL;DR

This paper establishes boundary Harnack inequalities, Carleson estimates, and growth properties for $p( abla)$-harmonic functions in domains with variable exponents, extending classical results to variable exponent settings.

Contribution

It proves the boundary Harnack inequality for $p( abla)$-harmonic functions with Lipschitz continuous exponents in $C^{1,1}$-domains, and develops Carleson estimates and measure growth results in NTA domains.

Findings

Boundary Harnack inequality for $p( abla)$-harmonic functions in Lipschitz domains.

Carleson estimates for $p( abla)$-harmonic functions in NTA domains.

Doubling property and growth estimates for $p( abla)$-harmonic measure.

Abstract

We investigate various boundary decay estimates for $p(\cdot)$-harmonic functions. For domains in $\mathbb{R}^n, n\geq 2$ satisfying the ball condition ($C^{1,1}$-domains) we show the boundary Harnack inequality for $p(\cdot)$-harmonic functions under the assumption that the variable exponent $p$ is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson type estimate for $p(\cdot)$-harmonic functions in NTA domains in $\mathbb{R}^n$ and provide lower- and upper- growth estimates and a doubling property for a $p(\cdot)$-harmonic measure.