p-harmonic functions by way of intrinsic mean value properties

Let Ω be a bounded domain in Rn . Under appropriate conditions on Ω, we prove existence and uniqueness of continuous functions solving the Dirichlet problem associated to certain nonlinear mean value properties in Ω with respect to balls of variable radius. We also show that, when properly normalized, such functions converge to the p-harmonic solution of the Dirichlet problem in Ω for p⩾2 . Existence is obtained via iteration, a fundamental tool being the construction of explicit universal barriers in Ω

p-harmonic functions by way of intrinsic mean value properties Articles