Short time behaviour for game-theoretic p-caloric functions

2017/09/28 by Diego Berti, Rolando Magnanini, Berti, Diego +1 · 1 citation
Computer Science · Mathematics · #35B40 #35K20 #35K92 #35Q91 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1709.10005

openalex publication_date 2017/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the solution of u_t-Δ^G_p u=0 in a (not necessarily bounded) domain, satisfying u=0 initially and u=1 on the boundary at all times. Here, Δ^G_p u is the game-theoretic or normalized p-laplacian. We derive new precise asymptotic formulas for short times, that generalize the work of S. R. S. Varadhan for large deviations and that of the second author and S. Sakaguchi for the heat content of a ball touching the boundary. We also compute the short-time behavior of the q-mean of u on such a ball. Applications to time-invariant level surfaces of u are then derived.

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