Electronic Journal of Differential Equations (Apr 2014)
Existence of solutions to a normalized F-infinity Laplacian equation
Abstract
In this article, for a continuous function F that is twice differentiable at a point $x_0$, we define the normalized F-infinity Laplacian $\Delta_{F; \infty}^N$ which is a generalization of the usual normalized infinity Laplacian. Then for a bounded domain $\Omega\subset\mathbb{R}^n$, $f\in C(\Omega)$ with $\inf_\Omega f(x)>0$ and $g\in C(\partial\Omega)$, we obtain existence and uniqueness of viscosity solutions to the Dirichlet boundary-value problem $$\displaylines{ \Delta_{F; \infty}^N u=f, \quad \text{in }\Omega,\cr u=g, \quad \text{on }\partial\Omega. }$$
