Existence of solutions to a normalized F-infinity Laplacian equation

• Main Authors: Hua Wang, Yijun He
• Format: Article
• Language: English
• Published: Texas State University 2014-04-01
• Series: Electronic Journal of Differential Equations
• Subjects: March 18, 2014. Published April 16, 2014
• Online Access: http://ejde.math.txstate.edu/Volumes/2014/109/abstr.html

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 $$\displaylines{ \Delta_{F; \infty}^N u=f, \quad \text{in }\Omega,\cr u=g, \quad \text{on }\partial\Omega. }$$