Asymptotics for the resolvent equation associated to the game-theoreticp-laplacian

Abstract

We consider the (viscosity) solution u ε of the elliptic equation ε 2 ∆ G p u = u in a domain (not necessarily bounded), satisfying u = 1 on its boundary. Here, ∆ G p is the game-theoretic or normalized p-laplacian. We derive asymptotic formulas for ε → 0 + involving the values of u ε , in the spirit of Varadhan's work [Va], and its q-mean on balls touching the boundary, thus generalizing that obtained in [MS1] for p = q = 2. As in a related parabolic problem, investigated in [BM], we link the relevant asymptotic behavior to the geometry of the domain. 2010 Mathematics Subject Classification. Primary 35J92; Secondary 35J25, 35B40, 35Q91.

Variational approach to the asymptotic mean-value property for the p-Laplacian on Carnot groups

“…Let Ω ⊂ G be an open neighborhood of 0 and let u ∈ C ∞ (Ω). Then, the following Taylor formula holds for any point P = (x (1) , x (2) , . .…”

“…where (x −1 y) (1) and (x −1 y) (2) are the horizontal and the vertical components of x −1 y, respectively and •, • R v 1 and •, • R v 2 denote the Euclidean scalar products on R v1 and R v2 , respectively. It then follows that…”

“…A new approach to the asymptotic mean-value property has been recently proposed in [11] (see also [2] for relations with statistical games). More precisely, in [11], the authors proved that every viscosity solution u to the normalized p-laplacian in an open set Ω ⊂ R n for a given 1 ≤ p ≤ ∞ (Definition 2.2), can be characterized using an asymptotic mean-value property in terms of the function µ p (ε, u)(x), defined as the unique minimizer of the following variational problem…”

“…Let G be a Carnot group of step k (Definition 2.1). Denote by ∆ N p,G the subelliptic normalized p-Laplacian (see (2) and ( 3)) and by µ p (ε, u) the generalized median of a function u defined uniquely as in (5). The theorem below stays that a viscosity solution of ∆ N p,G u = 0 can be characterized asymptotically by the minimum µ p (ε, u).…”

Variational approach to the asymptotic mean-value property for the p-Laplacian on Carnot groups

“…We note that ´B η, y (2) |y 1 | p−2 = 0, which follows by applying the change of variables ψ(y (1) , y (2) , y (3) , . .…”