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

TL;DR

The paper addresses the regularity of minimizers for the one-dimensional Cartesian area with a linear fidelity term to a small datum $g$, proving $C^{1,1}$ regularity in $I$ for minimizers in $BV(I)$. The approach combines anisotropic BV theory, Wulff-ball comparisons, and calibration-type arguments to relate subgraph regularity to the graph itself, establishing Lipschitz and, under ellipticity, $C^{1,1}$ regularity. Key contributions include a precise small-data regime ensuring regularity, a classification of local minimizers via Cahn–Hoffman vectors, and an extension to anisotropic settings with $L^p$ fidelity terms. The results partially confirm De Giorgi’s conjecture in dimension one and codimension one, with explicit dependence of the small-data threshold on the interval length and anisotropy, and have implications for variational models involving area-type functionals and BV regularity.

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.

Figures (3)

• Figure 1:
• Figure 2:
• Figure 3: The graph of $u$ in (a) and convex/concave nondecreasing envelope of $v$ in (b).

Theorems & Definitions (36)

• Conjecture 1.1
• Theorem 1.2
• Definition 2.1: Elliptic anisotropy
• Proposition 2.2
• Definition 2.3
• Lemma 2.4: Dalmaso:1980
• Definition 2.5
• Remark 2.6
• Proposition 2.7
• proof