Bounded variation of gradients in the Rudin–Osher–Fatemi denoising model
Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be a bounded smooth domain and $f \in C^\infty(\overline{\Omega})$. Let $U$ be the unique minimizer over $BV(\Omega) \cap L^2(\Omega)$ of the Rudin–Osher–Fatemi functional $$\mathcal{E}(u) = |Du|(\Omega) + \frac{1}{2} \int_\Omega |u - f|^2 \, dx,$$ where $|Du|$ is the total variation of the distributional gradient; no boundary values are prescribed. Caselles, Chambolle and Novaga proved that $\nabla U$ is locally bounded when $d \le 7$ and $\nabla f$ is locally bounded, and globally bounded when moreover $\Omega$ is convex; Porretta proved that $U$ is Lipschitz on $\Omega$ whenever $f$ is, in every dimension and on every smooth domain. In general $\nabla U$ is not continuous. For $\Omega = (0,1)$, Brezis proved that $f' \in BV(0,1)$ implies $U' \in BV(0,1)$.
Problem. Must $\nabla U \in BV(\Omega; \mathbb{R}^d)$? Equivalently, must all distributional second derivatives of $U$ be finite Radon measures on $\Omega$?
Source: Brezis (2019), Open Problem 1, who also asks whether at least ΔU is a measure; restated in Brezis (2023), Open Problem 8.1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, functions of bounded variation, total variation minimizers
References
@article{brezis2019,
author = {Brezis, Ha{\"i}m},
title = {Remarks on some minimization problems associated with {BV} norms},
journal = {Discrete Contin. Dyn. Syst.},
volume = {39},
number = {12},
year = {2019},
pages = {7013--7029},
doi = {10.3934/dcds.2019242},
}
@article{brezis2023,
author = {Brezis, Ha{\"i}m},
title = {Some of my favorite open problems},
journal = {Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.},
volume = {34},
number = {2},
year = {2023},
pages = {307--335},
doi = {10.4171/RLM/1008},
}
@article{rudin1992,
author = {Rudin, Leonid I. and Osher, Stanley and Fatemi, Emad},
title = {Nonlinear total variation based noise removal algorithms},
journal = {Phys. D},
volume = {60},
number = {1-4},
year = {1992},
pages = {259--268},
doi = {10.1016/0167-2789(92)90242-F},
}
@article{caselles2011,
author = {Caselles, Vicent and Chambolle, Antonin and Novaga, Matteo},
title = {Regularity for solutions of the total variation denoising problem},
journal = {Rev. Mat. Iberoam.},
volume = {27},
number = {1},
year = {2011},
pages = {233--252},
doi = {10.4171/RMI/634},
}
@article{porretta2021,
author = {Porretta, Alessio},
title = {On the regularity of the total variation minimizers},
journal = {Commun. Contemp. Math.},
volume = {23},
number = {1},
year = {2021},
pages = {1950082},
doi = {10.1142/S0219199719500822},
eprint = {1909.01695},
archivePrefix = {arXiv},
}
@misc{aim-rof-gradient-bounded-variation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Bounded variation of gradients in the Rudin–Osher–Fatemi denoising model}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/437-rof-gradient-bounded-variation.md},
note = {Entry 437 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00638, "Bounded variation of gradients in the Rudin–Osher–Fatemi denoising model", clopen.solutions.