On the existence and uniqueness of $p$-harmonious functions

Tug-of-war with Kolmogorov

“…We would like to thank an anonymous referee for emphasizing the subtle measure theoretical issues underlying the analysis in Section 4 and for encouraging us to reread [6]. This helped us to improve the analysis in Section 4.…”

“….. We claim that {v i } converges in a finite number of steps. To establish this, we argue as in the proof of Theorem 5.2 in [6] and use induction to show that…”

“…is Borel measurable for fixed (Y, t). Lemma 3.1 in [6] shows that for any fixed (Y, t) and any λ > 0, one can pick a Borel measurable map S Y :…”

“…Note that a subtle point here is that sup X∈B (X) and inf X∈B (X) are used instead of sup X∈B (X) and inf X∈B (X) in the definition of (p, )-Kolmogorov functions. In fact, if B (X) is used instead of B (X), then the Borel measurability of the functions in (4.7) can fail already for Y and t independent functions, see Example 2.4 in[6]. Moreover, due to the boundedness of v and the continuity in Y , one can see that is continuous and thus Borel measurable.…”

p-harmonic functions by way of intrinsic mean value properties

“…Furthermore, u ε → u uniformly in Ω as ε → 0, where u is the unique p-harmonic function solving the Dirichlet problem in Ω with boundary data f . It should be remarked that domains satisfying a uniform exterior cone condition (see Definition 1.4 below) verify the boundary regularity condition, in the sense of [20]; see also [1,8,18] for further approaches.…”