A mixed Sobolev estimate for reconstructing a scalar potential
Let $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$, with unit volume, and use the norms $$\|u\|_{H^s}^2 = \sum_{k \in \mathbb{Z}^d} (1 + 4\pi^2|k|^2)^s\, |\widehat{u}(k)|^2, \qquad \|u\|_{X+Y} = \inf_{u = a + b} \bigl(\|a\|_X + \|b\|_Y\bigr),$$ with $W^{1,1}$ carrying its usual norm; vector-valued norms use the Euclidean length and the sum of the component Fourier energies. Bourgain and Brezis showed that $\operatorname{div} Y = f$ has a solution $Y \in W^{1,d} \cap L^\infty$ for every $f \in L^d(\mathbb{T}^d)$ of mean zero, with norm bounds, which by duality gives $\|\psi - \textstyle\int \psi\|_{L^{d/(d-1)}} \le C_d \|\nabla\psi\|_{H^{-1/2} + L^1}$. They also proved $\|\varphi\|_{H^{1/2} + W^{1,1}} \le C(d)\bigl(1 + \|g\|_{H^{1/2}}\bigr)\|g\|_{H^{1/2}}$ for smooth real $\varphi$ and $g = e^{i\varphi}$, and observed that the estimate below is what this argument by duality would require. Both $H^{1/2}$ and $W^{1,1}$ embed in $L^{d/(d-1)}$.
Problem. For every integer $d \ge 2$, is there $C_d < \infty$ such that every real $\psi \in C^\infty(\mathbb{T}^d)$ with $\int_{\mathbb{T}^d} \psi = 0$ satisfies $$\|\psi\|_{H^{1/2}(\mathbb{T}^d) + W^{1,1}(\mathbb{T}^d)} \le C_d \|\nabla\psi\|_{H^{-1/2}(\mathbb{T}^d;\mathbb{R}^d) + L^1(\mathbb{T}^d;\mathbb{R}^d)}?$$
Source: Bourgain and Brezis (2003), (1.16); Brezis (2023), Open Problem 7.1. Stated with explicit norms on the unit torus by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- FA
- Keywords
- AIM Open Applied Problems, Bourgain–Brezis estimates, divergence equation, fractional Sobolev spaces
References
@article{bourgain2003,
author = {Bourgain, Jean and Brezis, Ha{\"\i}m},
title = {On the equation $\operatorname{div} {Y} = f$ and application to control of phases},
journal = {J. Amer. Math. Soc.},
volume = {16},
number = {2},
year = {2003},
pages = {393--426},
doi = {10.1090/S0894-0347-02-00411-3},
}
@article{brezis2023,
author = {Brezis, Ha{\"\i}m},
title = {Some of my favorite open problems},
journal = {Rend. Lincei Mat. Appl.},
volume = {34},
number = {2},
year = {2023},
pages = {307--335},
doi = {10.4171/RLM/1008},
}
@misc{aim-bourgain-brezis-mixed-sobolev-gradient-estimate,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A mixed Sobolev estimate for reconstructing a scalar potential}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/451-bourgain-brezis-mixed-sobolev-gradient-estimate.md},
note = {Entry 451 at commit a153f46, 2026-10-13},
}
Resolution claims
We construct smooth real mean-zero functions on the two-dimensional unit torus whose gradients stay bounded in $H^{-1/2}+L^1$, while their $H^{1/2}+W^{1,1}$ norms tend to infinity. This disproves the mixed potential estimate proposed by Bourgain and Brezis. The construction uses a finite stationary reflected walk to produce a bounded vector field with a uniformly bounded $H^{-1/2}$ divergence. Pairing this divergence with a …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00741, "A mixed Sobolev estimate for reconstructing a scalar potential", clopen.solutions.