The Perron method for solving the Dirichlet problem for $$p$$ p -harmonic functions is extended to unbounded open sets in the setting of a complete metric space with a doubling measure supporting a $$p$$ p -Poincaré inequality, $$1<p<\infty $$ 1 < p < ∞ . The upper and lower ( $$p$$ p -harmonic) Perron solutions are studied for open sets, which are assumed to be $$p$$ p -parabolic if unbounded. It is shown that continuous functions and quasicontinuous Dirichlet functions are resolutive (i.e., that their upper and lower Perron solutions coincide), that the Perron solution agrees with the $$p$$ p -harmonic extension, and that Perron solutions are invariant under perturbation of the function on a set of capacity zero.
Mathematische Zeitschrift – Springer Journals
Published: Mar 30, 2017
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