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

Diego Berti; Rolando Magnanini

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arXiv · 1801.04230

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

Diego Berti·Rolando Magnanini

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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.

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Diego Berti, Rolando Magnanini. 2018-01-12. Asymptotics for the resolvent equation associated to the game-theoretic $p$-laplacian. https://arxiv.org/abs/1801.04230

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