Asymptotics for the resolvent equation associated to the game-theoretic $p$-laplacian

TL;DR

This paper derives asymptotic formulas for solutions to a game-theoretic p-Laplacian equation as a parameter approaches zero, linking the behavior to domain geometry and generalizing previous results.

Contribution

It extends Varadhan's asymptotic analysis to the normalized p-Laplacian, providing new formulas involving solution values and boundary ball averages.

Findings

Asymptotic formulas for $ o 0^+$ involving solution values.

Generalization of previous results for $p=q=2$ to the game-theoretic p-Laplacian.

Connection between asymptotic behavior and domain geometry.

Abstract

We consider the (viscosity) solution $u^\varepsilon$ of the elliptic equation $\varepsilon^2\Delta_p^G u= u$ in a domain (not necessarily bounded), satisfying $u=1$ on its boundary. Here, $\Delta_p^G$ is the {\it game-theoretic or normalized $p$-laplacian}. We derive asymptotic formulas for $\varepsilon\to 0^+$ involving the values of $u^\varepsilon$, in the spirit of Varadhan's work \cite{Va}, and its $q$-mean on balls touching the boundary, thus generalizing that obtained in \cite{MS-AM} for $p=q=2$. As in a related parabolic problem, investigated in \cite{BM}, we link the relevant asymptotic behavior to the geometry of the domain.