Asymptotic analysis of solutions related to the game-theoretic p-laplacian

Abstract

We consider the (viscosity) solution $u(x,t)$ of the nonlinear evolution equation $u_t-\Delta^G_p u=0$ in a (not necessarily bounded) domain $\Omega$, such that $u=0$ in $\Omega$ at time $t=0$ and $u=1$ on the boundary of $\Omega$ at all times. Here, $\Delta_p^G$ is the game-theoretic $p$-laplacian, a $1$-homogeneous version of the standard $p$-laplacian. Also, we consider the (viscosity) solution $u^\varepsilon$ of the nonlinear elliptic equation $\varepsilon^2\Delta_p^G u^\varepsilon= u^\varepsilon$ in $\Omega$, satisfying $u^\varepsilon=1$ on its boundary. In this thesis, we establish asymptotic formulas for small positive values of $t$ and $\varepsilon$ involving both the values of $u(x,t)$ and $u^\varepsilon(x)$ and their $q$-means on balls touching the boundary. In the spirit of S.~R.~S.~Varadhan's work, we associate appropriate rescalings of the values of $u(x,t)$ and $u^\varepsilon(x)$ to the distance of $x$ to the boundary of $\Omega$. We also provide accurate uniform estimates of the rate of approximation in these formulas, highlighting the dependence on both the parameter $p$ and the regularity of the domain. The uniform estimates are new results also in the linear case. Also, we connect the asymptotic behavior of $q$-means on balls touching the boundary to a suitable function of principal curvatures. These results generalize and extend formulas for the heat content, obtained by R. Magnanini and S. Sakaguchi for $p=q=2$. Finally, we give a few applications of the asymptotic formulas to geometric and symmetry results. In particular, we characterize time-invariant level surfaces of $u(x,t)$ (or $\varepsilon$-invariant level surfaces of $u^\varepsilon(x)$) as spheres and hyperplanes.