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

Abstract

Abstract We prove a $$C^{1,1}$$ C 1 , 1 -regularity result for minimizers of the functional $$ \int _I \sqrt{1+|Du|^2} + \int _I |u-g|ds,\quad u\in BV(I), $$ ∫ I 1 + | D u | 2 + ∫ I | u - g | d s , u ∈ B V ( I ) , provided $$I\subset \mathbb {R}$$ I ⊂ R is a bounded open interval and $$\Vert g\Vert _\infty $$ ‖ g ‖ ∞ 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.