A De Giorgi conjecture on the regularity of minimizers of Cartesian area in 1D

We prove a $C^{1,1}$-regularity of minimizers of the functional $${\int }_I\sqrt{1+\mathrm{\mid }Du{\mathrm{\mid }}^2}+{\int }_I\mathrm{\mid }u-g\mathrm{\mid }ds,u\in BV(I),$$ provided $I\subset \mathbb{R}$ is a bounded open interval and $\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}$ is sufficiently small, thus partially establishing a De Giorgi conjecture in dimension one and codimension one. We also extend our result to a suitable anisotropic setting.