Preprint

Inserted: 27 jun 2026
Last Updated: 27 jun 2026

Year: 2026

Abstract:

We study the regularity of minimizers of one-dimensional convex functionals $\overline{\mathcal{F}}(u):BV(I;\mathbb{R}^k)\to\mathbb{R}$ with linear growth with respect to $u'$ and an $L^1$ fidelity term $\displaystyle \int_I \
u-w\
\ dx$. We prove that minimizers are of class $C^1(I;\mathbb{R}^k)$ whenever $w$ is sufficiently small either in $L^{\infty}$ or in $L^1$, with thresholds independent of the length of the interval. In the case of the relaxed length functional, this provides a positive answer to De Giorgi's conjecture \cite{DG} under the weaker assumption of smallness in the $L^1$ norm.

Keywords: relaxation, $\Gamma$-convergence, convexity, regularity of minimizers, approximation

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• DG_Conjecture.pdf