Bounded harmonic lifting for complex media
Let $d \ge 2$, let $\Omega \subset \mathbb{R}^d$ be a bounded connected $C^\infty$ domain, and let $C \in \mathbb{C}^{d \times d}$ be a constant matrix with $$\operatorname{Re} \sum_{j,k} C_{jk}\, \xi_k \overline{\xi_j} \ge \mu |\xi|^2 \qquad (\xi \in \mathbb{C}^d)$$ for some $\mu > 0$. For $\varphi \in H^{1/2}(\partial\Omega) \cap L^\infty(\partial\Omega)$ let $u_\varphi \in H^1(\Omega)$ be the function with trace $\varphi$ such that $$\int_\Omega C \nabla u_\varphi \cdot \overline{\nabla v} = 0 \qquad (v \in H^1_0(\Omega)),$$ the harmonic lifting of $\varphi$. Ter Elst and Ouhabaz proved, for Hölder continuous complex coefficients on domains of class $C^{1+\kappa}$, that the harmonic lifting extends to a bounded map from $L^p(\partial\Omega)$ to $L^p(\Omega)$ for every $1 \le p < \infty$, and from $C^\nu(\partial\Omega)$ to $C^\nu(\Omega)$ for $0 < \nu < \kappa$.
Problem. Does there always exist a finite constant $K(\Omega, C)$ such that $$\|u_\varphi\|_{L^\infty(\Omega)} \le K(\Omega, C)\, \|\varphi\|_{L^\infty(\partial\Omega)}$$ for every such $\varphi$?
Source: ter Elst and Ouhabaz (2025), Question 1, its constant-coefficient case on smooth domains. Stated by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- FA
- Keywords
- AIM Open Applied Problems, complex elliptic operators, harmonic lifting, maximum principle
References
@article{terelst2025five,
author = {ter Elst, A. F. M. and Ouhabaz, E. M.},
title = {Five open problems on the {D}irichlet-to-{N}eumann operator with variable coefficients},
journal = {Integral Equations Operator Theory},
volume = {97},
number = {2},
year = {2025},
pages = {14},
doi = {10.1007/s00020-025-02796-9},
}
@article{terelst2025commutator,
author = {ter Elst, A. F. M. and Ouhabaz, E. M.},
title = {Commutator estimates and {P}oisson bounds for {D}irichlet-to-{N}eumann operators with variable coefficients},
journal = {Calc. Var. Partial Differential Equations},
volume = {64},
number = {2},
year = {2025},
pages = {58},
doi = {10.1007/s00526-024-02899-y},
eprint = {2404.18272},
archivePrefix = {arXiv},
}
@misc{aim-complex-harmonic-lifting,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Bounded harmonic lifting for complex media}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/654-complex-harmonic-lifting.md},
note = {Entry 654 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
Yes: for every d >= 2, every bounded connected C^infty domain Omega and every constant accretive C there is K(Omega, C) < infinity with ||u_phi||_inf <= K ||phi||_inf for all phi in H^{1/2} cap L^inf, from Agmon's 1960 maximum estimate with its L^1 remainder controlled by an L^2 duality bound, extended from smooth data by heat smoothing on the boundary..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 218-complex-harmonic-lifting/PROOF.md. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00670, "Bounded harmonic lifting for complex media", clopen.solutions.