ON DIRICHLET PROBLEMS INVOLVING THE INFINITY-LAPLACIAN

Abstract: We discuss viscosity solutions $u\in C(\overline{\Omega})$ to the Dirichlet problem \[ (DP)\qquad \begin{cases} \Delta_{\infty}u=f(x,u), & x\in\Omega,\\ u=b, & x\in\partial\Omega. \end{cases} \] Here $\Omega\subset\mathbb{R}^{n}$ is a bounded domain, $b\in\partial\Omega$, $f$ is continuous on $\Omega\times\mathbb{R}$ and $\Delta_{\infty}u$ is the infinity Laplacian, a highly degenerate elliptic operator, given by $\Delta_{\infty}u=\langle D^{2}uDu,Du\rangle$. General conditions on $f$ under which the above Dirichlet problem admits a solution in $C(\overline{\Omega})$ for any bounded domain $\Omega$ will be given. In contrast, we will identify a class of inhomogeneous terms $f$ for which Problem (DP) has no solution in $C(\overline{\Omega})$ provided $\Omega$ contains a large ball. Some Comparison principles and Harnack inequalities will also be discussed. Finally we will mention several open problems. This talk is based on a joint work with Tilak Bhattacharya.