Summary: Let \(\Omega\) be a domain of \(\mathbb{R}^N\), \(N \geq 2\), with non empty boundary \(\Gamma\). In these notes, we deal with the solution \(u\) of \(u_t = F\left (\nabla u, \nabla^2 u\right)\) in \(\Omega \times (0, \infty)\), such that \(u\) is initially zero in \(\Omega\) and equals one on \(\Gamma\) for all positive times. Here, \(F\) is the game-theoretic \(p\)-Laplacian \(\Delta_p^G\) or either one of the Pucci’s extremal operators \(\mathcal{M}^\pm\). In the spirit of works by Varadhan and Magnanini-Sakaguchi in the case of the same initial-boundary problem for the heat equation, we summarize recent results regarding the connection between the behavior for small times and the geometry of \(\Omega\). In particular, we present asymptotic formulas as \(t \rightarrow 0^+\) for both the values of \(u\) and of its \(q\)-means on balls touching \(\Gamma\).
For the entire collection see [Zbl 1497.42002].