A discrete stochastic interpretation of the dominative \(p\)-Laplacian. (English) Zbl 1474.35292

Brustad, Karl K.; Lindqvist, Peter; Manfredi, Juan J.

Differ. Integral Equ. 33, No. 9-10, 465-488 (2020).

Authors’ abstract: We build a discrete stochastic process adapted to the (nonlinear) dominative \(p\)-Laplacian \[\mathcal{D}_pu(x):=\Delta u+(p-2)\lambda_{N},\] where \(\lambda_{N}\) is the largest eigenvalue of \(D^2u\) and \(p>2\). We show that the discrete solutions of the Dirichlet problems at scale \(\varepsilon\) tend to the solution of the Dirichlet problem for \(\mathcal{D}_p\) as \(\varepsilon\to 0\). We assume that the domain and the boundary values are both Lipschitz.

Reviewer: Stepan Agop Tersian (Rousse)

MSC:

35J60	Nonlinear elliptic equations
35J92	Quasilinear elliptic equations with \(p\)-Laplacian
49L20	Dynamic programming in optimal control and differential games
91A15	Stochastic games, stochastic differential games

Keywords:

stochastic game theory; dominative \(p\)-Laplace equation; non-linear mean value operator