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

Abstract

We prove a \(C^{1,1}\)-regularity result for minimizers of the functional

$$ \int _I \sqrt{1+|Du|^2} + \int _I |u-g|ds,\quad u\in BV(I), $$

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.