Analytic boundary diffusion on continuous functions for Lipschitz domains
Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be a bounded connected Lipschitz domain, and let $C$ be a real symmetric, uniformly positive definite matrix field with Lipschitz entries on $\overline{\Omega}$. Let $N$ be the weak Dirichlet-to-Neumann operator for $-\operatorname{div}(C\nabla)$, defined by harmonic extension and conormal differentiation. Arendt and ter Elst proved that the semigroup generated by $-N$ leaves $C(\partial\Omega)$ invariant and restricts to a strongly continuous semigroup $T(t)$ on $C(\partial\Omega)$ with the supremum norm. Under more boundary smoothness, a $C^{1+\kappa}$-boundary for some $\kappa > 0$ with Hölder continuous coefficients, ter Elst and Ouhabaz proved that this semigroup on $C(\partial\Omega)$ is holomorphic with angle $\pi/2$.
Problem. Must there exist $\theta > 0$ such that $T$ extends to an operator-valued holomorphic semigroup on $\{z \ne 0 : |\arg z| < \theta\}$, with $$\sup_{0 < |z| \le 1,\ |\arg z| \le \theta'} \|T(z)\| < \infty$$ for each $0 < \theta' < \theta$?
Source: ter Elst and Ouhabaz (2025), Question 4, final paragraph; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, Dirichlet-to-Neumann operator, holomorphic semigroups, Lipschitz domains
References
@article{terelst2025,
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{arendt2020,
author = {Arendt, Wolfgang and ter Elst, A. F. M.},
title = {The {D}irichlet-to-{N}eumann operator on {$C(\partial\Omega)$}},
journal = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)},
volume = {20},
year = {2020},
pages = {1169--1196},
doi = {10.2422/2036-2145.201708_010},
eprint = {1707.05556},
archivePrefix = {arXiv},
}
@article{terelst2019,
author = {ter Elst, A. F. M. and Ouhabaz, E. M.},
title = {Analyticity of the {D}irichlet-to-{N}eumann semigroup on continuous functions},
journal = {J. Evol. Equ.},
volume = {19},
number = {1},
year = {2019},
pages = {21--31},
doi = {10.1007/s00028-018-0467-x},
eprint = {1707.07718},
archivePrefix = {arXiv},
}
@misc{aim-dtn-analyticity-continuous-boundary,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Analytic boundary diffusion on continuous functions for Lipschitz domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/431-dtn-analyticity-continuous-boundary.md},
note = {Entry 431 at commit a153f46, 2026-10-13},
}
Resolution claims
Let $\Omega\subset\R^d$, $d\geq2$, be a bounded connected Lipschitz domain, and let $C$ be a real symmetric uniformly elliptic matrix with Lipschitz entries on $\overline\Omega$. We prove that the weak Dirichlet-to-Neumann semigroup on $C(\partial\Omega)$ is holomorphic of angle $\pi/2$ and bounded on every strictly smaller sector. This answers the continuous-function question in the real symmetric case of Question 4 of ter …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, paper/dtn_analyticity.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00744, "Analytic boundary diffusion on continuous functions for Lipschitz domains", clopen.solutions.