Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 36, 2011, 261-278

THE BOUNDARY HARNACK INEQUALITY FOR SOLUTIONS TO EQUATIONS OF ARONSSON TYPE IN THE PLANE

Niklas L.P. Lundström and Kaj Nyström

Umeå University, Department of Mathematics and Mathematical Statistics
S-901 87 Umeå, Sweden; niklas.lundstrom 'at' math.umu.se

Umeå University, Department of Mathematics and Mathematical Statistics
S-901 87 Umeå, Sweden; kaj.nystrom 'at' math.umu.se

Abstract. In this paper we prove a boundary Harnack inequality for positive functions which vanish continuously on a portion of the boundary of a bounded domain \Omega \subset R² and which are solutions to a general equation of p-Laplace type, 1 < p < \infty. We also establish the same type of result for solutions to the Aronsson type equation \nabla (F(x,\nabla u)) \cdot F_\eta(x,\nabla u) = 0. Concerning \Omega we only assume that \partial\Omega is a quasicircle. In particular, our results generalize the boundary Harnack inequalities in [BL] and [LN2] to operators with variable coefficients.

2000 Mathematics Subject Classification: Primary 35J25, 35J70.

Key words: Boundary Harnack inequality, p-Laplace, A-harmonic function, infinity harmonic function, Aronsson type equation, quasicircle.

Reference to this article: N.L.P. Lundström and K. Nyström: The boundary Harnack inequality for solutions to equations of Aronsson type in the plane. Ann. Acad. Sci. Fenn. Math. 36 (2011), 261-278.

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doi:10.5186/aasfm.2011.3616