Ricerc@Sapienza

We prove two rigidity results for a variational infinity ground state $u$ of an open bounded convex domain $Omega subset R ^n$.
They state that $u$ coincides with a multiple of the distance from the boundary of $Omega$ if either $|
abla u|$ is constant on $partial Omega$, or $u$ is of class $C ^ {1,1}$ outside the high ridge of $Omega$. Consequently, in both cases $Omega$ can be geometrically characterized as a ``stadium-like domain''.

Aim of this note is to study the infinity Laplace operator and the corresponding Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket in the spirit of the classical construction of Kigami for the Laplacian. We introduce a notion of infinity harmonic functions on pre-fractal sets and we show that these functions solve a Lipschitz extension problem in the discrete setting. Then we prove that the limit of the infinity harmonic functions on the pre-fractal sets solves the Absolutely Minimizing Lipschitz Extension problem on the Sierpinski gasket.