On the definition and properties of $p$-harmonious functions

Abstract

We consider functions that satisfy the identityHere ε > 0 and α, and β are suitable nonnegative coefficients such that α + β = 1. In particular, we show that these functions are uniquely determined by their boundary values, approximate p-harmonic functions for 2 ≤ p < ∞ (for a choice of p that depends on α and β), and satisfy the strong comparison principle. We also analyze their relation to the theory of tug-of-war games with noise.

Convergence of dynamic programming principles for the p-Laplacian

“…Next we prove Theorem 2.11, i.e. we present the proof of convergence for Lipschitz domains following the proof sketched in [MPR12] in the case of p-harmonious functions. See Figure 1 below for a graphical explanation of the proof.…”

“…When p ≥ 2, α = p−2 p+d , and β = 2+d p+d , it turns out that there is always a unique solution to (1.2) that we label u ε and call (ε, p)-harmonious. Manfredi-Parviainen-Rossi proved in [MPR12] that u ε converges uniformly to u the solution of (1.1) when ε → 0. This theorem has been extended to the case 1 < p < 2 with variable p(x) ([AHP17]), to the single and double obstacle problems ([LM17, CLM17]), and to the Heisenberg group ( [LMR18]).…”

“…In the language of [BS91] we show the that p-Laplacian satisfies the "strong uniqueness property". For Lipschitz domains, we adapt a boundary regularity argument from [MPR12] showing the existence of good barriers near boundary points. These barriers are built on certain ring domains using the fundamental solution of the p-Laplacian with pole at the center of the ring, which are C 2 -domains.…”

Gradient and Lipschitz Estimates for Tug-of-War Type Games

“…In [MPR12] Manfredi, Parviainen and Rossi studied a variant of the tug-of-war game and its connection to the dynamic programming principle (DPP)…”

The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian

“…For more applications in image processing, one can see [11,12] etc. For more results about the normalized p-Laplacian and related topics about tug-of-war, one can see [19,20,21,27,28,29] and the references therein.…”