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

Abstract

We prove a $C^{1,1}$-regularity of minimizers of the functional $${\int }_I\sqrt{1+|Du{|}^2}+{\int }_I|u-g|ds,u\in BV(I),$$ provided $I\subset \mathbb{R}$ is a bounded open interval and $\Vert g{\Vert }_{\infty }$ 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.

19 pages, 3 figures