manuscripta math. 105, 401 – 423 (2001) © Springer-Verlag 2001 Juha Kinnunen · Nageswari Shanmugalingam Regularity of quasi-minimizers on metric spaces Received: 12 May 2000 / Revised version: 20 April 2001 Abstract. Using the theory of Sobolev spaces on a metric measure space we are able to apply calculus of variations and define p-harmonic functions as minimizers of the p-Dirichlet integral. More generally, we study regularity properties of quasi-minimizers of p-Dirichlet integrals in a metric measure space. Applying the De Giorgi method we show that quasi- minimizers, and in particular p-harmonic functions, satisfy Harnack’s inequality, the strong maximum principle, and are locally Hölder continuous, if the space is doubling and supports a Poincaré inequality. 1. Introduction The classical Dirichlet problem is to find a harmonic function with given boundary values. An alternative variational formulation of this problem is to minimize the Dirichlet integral  |Du| 2 dx among all functions which have required boundary values.A more general nonlinear variation of the classical Dirichlet problem is to study minimizers of the p-Dirichlet integral  |Du| p dx, with 1 <p< ∞. The minimizers are solutions to the corresponding Euler– Lagrange equation, which in this case is the p-Laplace equation div(|Du| p-2 Du) = 0, and continuous solutions are called p-harmonic functions. It is not clear what the counterpart for the p-Laplace equation is in a general metric measure space, but the variational approach is available; it is possible to J. Kinnunen: Institute of Mathematics, P.O. Box 1100, FIN-02015 Helsinki University of Technology, Finland. e-mail: juha.kinnunen@hut.fi N. Shanmugalingam: Department of Mathematics, University of Texas, Austin, TX 78712, USA. e-mail: nageswari@math.utexas.edu Mathematics Subject Classification (2000): 49N60, 35J60