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The boundary Harnack inequality for variable exponent $p$-Laplacian, Carleson estimates, barrier functions and $p(\cdot)$-harmonic measures (1405.2678v1)
Published 12 May 2014 in math.AP
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.