Extending infinity harmonic functions by rotation
Gustaf Gripenberg · DOAJ (DOAJ: Directory of Open Access Journals) · 2015
If $u(\mathbf{x}, y)$ is an infinity harmonic function, i.e., a viscosity solution to the equation $-\Delta_\infty u=0$ in $\Omega \subset \mathbb{R}^{m+1}$ then the function $v(\mathbf{x}, \mathbf{z})= u(\mathbf{x}, \|\mathbf{z}\|)$ is infinity harmonic in the set $\{(\mathbf{x}, \mathbf{z}): (\mathbf{x}, \|\mathbf{z}\|)\in \Omega\}$ (provided $u(\mathbf{x},-y)=u(\mathbf{x},y)$).