Asymptotics for the resolvent equation associated to the game-theoretic \(p\)-Laplacian. (English) Zbl 1421.35164
Summary: We consider the (viscosity) solution \(u^\varepsilon\) of the elliptic equation \(\varepsilon^2 \Delta^G_p u=u\) in a domain (not necessarily bounded), satisfying \(u=1\) on its boundary. Here, \(\Delta^G_p\) is the game-theoretic or normalized \(p\)-Laplacian. We derive asymptotic formulas for \(\varepsilon\to 0^+\) involving the values of \(u^\varepsilon\), in the spirit of S. R. S. Varadhan’s work [Commun. Pure Appl. Math. 20, 431–455 (1967; Zbl 0155.16503)], and its \(q\)-mean on balls touching the boundary, thus generalizing that obtained by the second author and S. Sakaguchi for \(p=q=2\). As in a related parabolic problem, investigated in a previous work by the authors [J. Math. Pures Appl. (9) 126, 249–272 (2019; Zbl 1412.35191)], we link the relevant asymptotic behavior to the geometry of the domain.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
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