Abstract
We study Harnack type properties of quasiminimizers of the \(p\mspace{1mu}\)-Dirichlet integral on metric measure spaces equipped with a doubling measure and supporting a Poincaré 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 \(p\mspace{1mu}\)-Dirichlet integral. In the Euclidean case with the Lebesgue measure we obtain convergence also in the Sobolev norm.
Keywords: Metric space, doubling measure, Poincaré inequality, Newtonian space, Harnack inequality, Harnack convergence theorem
Mathematics Subject Classification (2000): 49J52, 35J60, 49J27
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Kinnunen, J., Marola, N. & Martio, O. Harnack’s principle for quasiminimizers. Ricerche mat. 56, 73–88 (2007). https://doi.org/10.1007/s11587-007-0006-5
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DOI: https://doi.org/10.1007/s11587-007-0006-5