OP-00467 Open

Boundary-flux commutators for rough conductors

Note: AIM — Open Applied Problems marks this solved (15 October 2026) by a counterexample built from Meyers' half-disk example (N. G. Meyers, Ann. Scuola Norm. Sup. Pisa 17 (1963), 189–206), through a reduction made in AIM's AI review. The article is not on a site the registry links papers from, and the reduction has not been checked here.

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Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be a bounded connected Lipschitz domain, and let $C \in L^\infty(\Omega; \mathbb{R}^{d \times d})$ be symmetric with $\mu I \le C(x) \le \mu^{-1} I$ for almost every $x$, for some $\mu > 0$. For $\varphi \in H^{1/2}(\partial\Omega)$ let $u_\varphi \in H^1(\Omega)$ be the weak solution of $\operatorname{div}(C \nabla u) = 0$ in $\Omega$ with trace $\varphi$. The Dirichlet-to-Neumann operator $N$ in $L^2(\partial\Omega)$ is defined by $\varphi \in D(N)$ and $N\varphi = h$ if $h \in L^2(\partial\Omega)$ and $$\int_\Omega C \nabla u_\varphi \cdot \overline{\nabla v} = \int_{\partial\Omega} h \, \overline{\operatorname{Tr} v} \, dS \qquad (v \in H^1(\Omega)).$$ Shen proved the estimate below for Lipschitz $\varphi$ and $g$ on $\partial\Omega$ when $C$ is in addition Hölder continuous, with $K$ a multiple of the Lipschitz norm of $g$ on $\partial\Omega$. Ter Elst and Ouhabaz proved it for complex Hölder continuous coefficients on bounded $C^{1+\kappa}$ domains, and Hofmann and Zhang proved $L^2$ commutator estimates on the upper half-space for complex bounded measurable coefficients independent of the transversal variable.

Problem. For every $g \in C^\infty(\mathbb{R}^d)$, is there $K < \infty$ such that $g\varphi \in D(N)$ and $$\|N(g\varphi) - g N\varphi\|_{L^2(\partial\Omega)} \le K \|\varphi\|_{L^2(\partial\Omega)}$$ for every $\varphi \in D(N)$? Here $g$ on $\partial\Omega$ means its restriction, and $K$ may depend on $\Omega$, $C$ and $g$.

Source: ter Elst and Ouhabaz (2025), open problem 2, for complex bounded measurable coefficients; the case of real symmetric coefficients that need not be Hölder continuous, which they single out, is stated by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: AP — Analysis of PDEs
Secondary: CA, FA
Keywords: AIM Open Applied Problems, commutator estimates, Dirichlet-to-Neumann operators, rough coefficients

References

@article{terelst2025open,
  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   = {Paper No. 14},
  doi     = {10.1007/s00020-025-02796-9},
}
@incollection{shen2016,
  author        = {Shen, Zhongwei},
  title         = {Commutator estimates for the {D}irichlet-to-{N}eumann map in {L}ipschitz domains},
  booktitle     = {Some Topics in Harmonic Analysis and Applications},
  editor        = {Li, Ji and Li, Xiaochun and Lu, Guozhen},
  series        = {Advanced Lectures in Mathematics},
  volume        = {34},
  publisher     = {International Press and Higher Education Press},
  address       = {Somerville, MA and Beijing},
  year          = {2016},
  pages         = {369--384},
  eprint        = {1308.6226},
  archivePrefix = {arXiv},
}
@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         = {Paper No. 58},
  doi           = {10.1007/s00526-024-02899-y},
  eprint        = {2404.18272},
  archivePrefix = {arXiv},
}
@article{hofmann2021,
  author        = {Hofmann, Steve and Zhang, Guoming},
  title         = {{$L^2$} estimates for commutators of the {D}irichlet-to-{N}eumann map associated to elliptic operators with complex-valued bounded measurable coefficients on {$\mathbb{R}^{n+1}_+$}},
  journal       = {J. Math. Anal. Appl.},
  volume        = {504},
  number        = {2},
  year          = {2021},
  pages         = {125408},
  doi           = {10.1016/j.jmaa.2021.125408},
  eprint        = {2102.06835},
  archivePrefix = {arXiv},
}
@misc{aim-rough-coefficient-boundary-commutator,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Boundary-flux commutators for rough conductors}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/216-rough-coefficient-boundary-commutator.md},
  note         = {Entry 216 at commit a153f46, 2026-10-13},
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00467, "Boundary-flux commutators for rough conductors", clopen.solutions.