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arXiv · 2505.15586

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

Giovanni Bellettini·Shokhrukh Yu. Kholmatov
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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.

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Giovanni Bellettini, Shokhrukh Yu. Kholmatov. 2025-05-21. A De Giorgi conjecture on the regularity of minimizers of Cartesian area in 1D. https://arxiv.org/abs/2505.15586

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