Abstract
The Perron method for solving the Dirichlet problem for p-harmonic functions is extended to unbounded open sets in the setting of a complete metric space with a doubling measure supporting a p-Poincaré inequality, 1<p<infty . The upper and lower (p-harmonic) Perron solutions are studied for open sets, which are assumed to be 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-harmonic extension, and that Perron solutions are invariant under perturbation of the function on a set of capacity zero.
Highlights
The Dirichlet problem for p-harmonic functions, 1 < p < ∞, which is a nonlinear generalization of the classical Dirichlet problem, considers the p-Laplace equation, pu := div(|∇u|p−2∇u) = 0, (1.1)
The nonlinear potential theory of p-harmonic functions has been developed since the 1960s; in Rn, and in weighted Rn, Riemannian manifolds, and other settings
Björn–Björn–Shanmugalingam [7] extended the Perron method for p-harmonic functions to the setting of a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality, and proved that Perron solutions are p-harmonic and agree with the previously obtained solutions for Newtonian boundary data in Shanmugalingam [33]
Summary
The Dirichlet (boundary value) problem for p-harmonic functions, 1 < p < ∞, which is a nonlinear generalization of the classical Dirichlet problem, considers the p-Laplace equation, pu := div(|∇u|p−2∇u) = 0,. With prescribed boundary values u = f on the boundary ∂. A continuous weak solution of (1.1) is said to be p-harmonic. The nonlinear potential theory of p-harmonic functions has been developed since the 1960s; in Rn, and in weighted Rn, Riemannian manifolds, and other settings. The books Malý–Ziemer [28] and Heinonen–Kilpeläinen–Martio [18] are two thorough treatments in Rn and weighted Rn, respectively
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