Continuous harmonic extension of every degree-zero sphere boundary map
Let $B^3 \subset \mathbb{R}^3$ be the unit ball. A weakly harmonic map from $B^3$ to $S^2$ is a map $u \in H^1(B^3; \mathbb{R}^3)$ with $|u| = 1$ almost everywhere and $$-\Delta u = |\nabla u|^2 u \quad \text{in } B^3$$ in the sense of distributions. Let $g \in C^\infty(S^2; S^2)$ have topological degree zero; when the degree of $g$ is not zero, no continuous map $\overline{B^3} \to S^2$ extends $g$. Minimizers of the Dirichlet energy among maps in $H^1(B^3; S^2)$ with trace $g$ are smooth on $\overline{B^3}$ except at finitely many interior points (Schoen and Uhlenbeck; Brezis, Coron and Lieb). Hardt and Lin found smooth maps $g$ of degree zero for which this minimum is strictly smaller than the infimum of the energy over continuous maps with trace $g$, so that no minimizer is continuous.
Problem. For every such $g$, is there a weakly harmonic map $$u \in H^1(B^3; \mathbb{R}^3) \cap C(\overline{B^3}; S^2), \qquad u|_{S^2} = g?$$ The map is not required to minimize the Dirichlet energy.
Source: R. Schoen (mid-1980s); see Brezis (2023), Open Problem 4.1, after Hardt and Lin (1986) and Hardt (1997).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, boundary value problems, harmonic maps
References
@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},
}
@article{hardt1986,
author = {Hardt, Robert and Lin, Fang-Hua},
title = {A remark on {$H^1$} mappings},
journal = {Manuscripta Math.},
volume = {56},
number = {1},
year = {1986},
pages = {1--10},
doi = {10.1007/BF01171029},
}
@article{hardt1997,
author = {Hardt, Robert M.},
title = {Singularities of harmonic maps},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {34},
number = {1},
year = {1997},
pages = {15--34},
doi = {10.1090/S0273-0979-97-00692-7},
}
@article{schoen1982,
author = {Schoen, Richard and Uhlenbeck, Karen},
title = {A regularity theory for harmonic maps},
journal = {J. Differential Geom.},
volume = {17},
number = {2},
year = {1982},
pages = {307--335},
doi = {10.4310/jdg/1214436923},
}
@article{schoen1983,
author = {Schoen, Richard and Uhlenbeck, Karen},
title = {Boundary regularity and the {D}irichlet problem for harmonic maps},
journal = {J. Differential Geom.},
volume = {18},
number = {2},
year = {1983},
pages = {253--268},
doi = {10.4310/jdg/1214437663},
}
@article{brezis1986,
author = {Brezis, Ha{\"i}m and Coron, Jean-Michel and Lieb, Elliott H.},
title = {Harmonic maps with defects},
journal = {Comm. Math. Phys.},
volume = {107},
number = {4},
year = {1986},
pages = {649--705},
doi = {10.1007/BF01205490},
}
@misc{aim-degree-zero-harmonic-extension,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Continuous harmonic extension of every degree-zero sphere boundary map}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/435-degree-zero-harmonic-extension.md},
note = {Entry 435 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-00855, "Continuous harmonic extension of every degree-zero sphere boundary map", clopen.solutions.