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.
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Received: 12 May 2000 / Revised version: 20 April 2001
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Kinnunen, J., Shanmugalingam, N. Regularity of quasi-minimizers on metric spaces. manuscripta math. 105, 401–423 (2001). https://doi.org/10.1007/s002290100193
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DOI: https://doi.org/10.1007/s002290100193