On the Dirichlet problem for solutions of a restricted nonlinear mean value property

p-harmonic functions by way of intrinsic mean value properties

“…Theorem 1.5 below is an existence and uniqueness result for the Dirichlet problem associated to intrinsic mean value properties. It extends the existence result in [20] to the variable radius setting, substantially relaxes the geometrical restrictions of [3] and, as an additional feature, the solution is constructively obtained via iteration of the averaging operators T ρ,p (see (1.6)).…”

“…It is important to remark at this point that the main existence and convergence results (Theorems 1.5 and 1.7 below) of this paper require the admissible radius function to be continuous in Ω. However, this assumption, in spite of its essential role in the proof of the local equicontinuity [3,4], is not directly involved in the proofs presented in this paper, which are focused on the behavior of the iterates near the boundary (see Section 3).…”

“…We now state the first main result of this paper (compare with [3], where the same result was proven under the assumption that Ω is strictly convex and the admissible radius function is 1-Lipschitz); see [18,20] for previous versions in the constant radius case.…”

“…As for equicontinuity, it is worth mentioning that boundary equicontinuity turns out to be a much more delicate matter than interior equicontinuity. In [3], boundary equicontinuity was established under the assumption that Ω is strictly convex and ρ is 1-Lipschitz. Our approach here is based on the construction of explicit barriers for T ρ,p , having the additional advantage that they work for domains satisfying the uniform exterior cone condition.…”

“…We start with the following comparison principle that uses a standard argument (see also [3,Proposition 4.1]). Proof.…”