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

ABSTRACT

We consider the (viscosity) solution $u^{\epsilon }$ of the elliptic equation ${\epsilon }^2{\mathrm{\Delta }}_p^{G}u=u$ in a domain (not necessarily bounded), satisfying $u=1$ on its boundary. Here, ${\mathrm{\Delta }}_p^G$ is the game-theoretic or normalized p-laplacian. We derive asymptotic formulas for $\epsilon \to 0^{+}$ involving the values of $u^{\epsilon }$, in the spirit of Varadhan’s work, and its q-mean on balls touching the boundary, thus generalizing that obtained by R. Magnanini and S. Sakaguchi for $p=q=2$. As in a related parabolic problem, investigated in a previous work by the authors, we link the relevant asymptotic behavior to the geometry of the domain.

KEYWORDS:

• Game-theoretic p-laplacian
• asymptotic formulas
• q-means

AMS Subject Classifications:

• Primary: 35J92
• Secondary: 35J25
• 35B40
• 35Q91

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The paper was partially supported by the Gruppo Nazionale Analisi Matematica Probabilità e Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).