Harnack’s principle for quasiminimizers Juha Kinnunen Department of Mathematical Sciences, University of Oulu, P.O. Box 3000 FI-90014 University of Oulu, Finland ; juha.kinnunen@oulu.fi Niko Marola Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100 FI-02015 Helsinki University of Technology, Finland ; nmarola@math.hut.fi Olli Martio Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 FI-00014 University of Helsinki, Finland ; olli.martio@helsinki.fi Abstract. We study Harnack type properties of quasiminimizers of the p -Dirichlet integral on metric measure spaces equipped with a doubling measure and supporting a Poincar´ e inequality. We show that an increasing sequence of quasiminimizers converges locally uniformly to a quasiminimizer, provided the limit function is finite at some point, even if the quasiminimizing constant and the boundary values are allowed to vary in a bounded way. If the quasiminimizing constants converge to one, then the limit function is the unique minimizer of the p -Dirichlet integral. In the Euclidean case with the Lebesgue measure we obtain convergence also in the Sobolev norm. Key words and phrases : Metric space, doubling measure, Poincar´ e inequality, Newto- nian space, Harnack inequality, Harnack convergence theorem. Mathematics Subject Classification (2000): Primary: 49J52; Secondary: 35J60, 49J27. 1. Introduction Let Ω ⊂ R n be a bounded open set and 1 <p< ∞. A function u ∈ W 1,p loc (Ω) is a Q-quasiminimizer, Q ≥ 1, of the p -Dirichlet integral in Ω if for all open Ω ′ ⋐ Ω 1