How to show that $1$-laplacian $\Delta_1$ is elliptic?

Question

How to show that $\Delta_1$ operator defined as: \begin{equation} \Delta_1u=\text{div}\Bigg(\frac{\nabla u}{\|\nabla u\|}\Bigg) \end{equation} is elliptic ? I know that $\Delta_1 u$ is the E-L equation of the associated energy $\int_\Omega f(\nabla u)\:dx=\int_\Omega|\nabla u|\: dx$. I have seen the proof of ellipticity for $p$-laplacian ($\Delta_p$) here. But since the integrand $f$ in this case fails to be of class $C^2$ because of non-differentiability at origin, the weaker ellipticity condition (Legendre-Hadamard condition) \begin{equation} \sum_{i=1}^n\sum_{j=1}^n\frac{\partial^2 f}{\partial \xi_j\partial \xi_i}(\xi)\lambda_i\lambda_j\ge 0, \lambda\in\mathbb{R}^n \end{equation} cannot be verfied directly. How to proceed ?