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

TL;DR

This paper proves $C^{1,1}$ regularity of minimizers for a specific functional in one dimension, partially confirming a De Giorgi conjecture and extending results to anisotropic cases.

Contribution

It establishes the $C^{1,1}$ regularity of minimizers in 1D for the Cartesian area functional, advancing the understanding of the De Giorgi conjecture.

Findings

Minimizers are $C^{1,1}$ regular under small $g$.

Partial proof of De Giorgi conjecture in 1D.

Extension to anisotropic settings.

Abstract

We prove a $C^{1,1}$-regularity of 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 $\|g\|_\infty$ 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.