Poisson kernel bounds for elliptic boundary diffusion on Lipschitz domains
Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be a bounded connected Lipschitz domain with boundary $\Gamma = \partial\Omega$ and surface measure $\sigma$, and let $C(x)$ be a real symmetric matrix depending smoothly on $x$ in a neighbourhood of $\overline{\Omega}$, with $\xi^{\mathsf T} C(x) \xi \ge \mu |\xi|^2$ for some $\mu > 0$. The Dirichlet-to-Neumann operator $N$ on $L^2(\Gamma)$ is defined weakly: $\varphi \in D(N)$ and $N\varphi = \psi \in L^2(\Gamma)$ if some $u \in H^1(\Omega)$ with $\operatorname{div}(C \nabla u) = 0$ and $u|_\Gamma = \varphi$ satisfies $$\int_\Omega C\nabla u \cdot \overline{\nabla v} = \int_\Gamma \psi\, \overline{v|_\Gamma}\, d\sigma$$ for every $v \in H^1(\Omega)$, so that formally $N\varphi = \nu \cdot C\nabla u$. It generates the semigroup $e^{-tN}$ on $L^2(\Gamma)$. For the Laplacian, ter Elst and Ouhabaz proved Poisson bounds for its kernel when $\Gamma$ is $C^\infty$, and for real Hölder continuous coefficients they proved the bound below when $\Omega$ is of class $C^{1+\kappa}$ for some $\kappa > 0$.
Problem. Does $e^{-tN}$ have an integral kernel $K_t$, with respect to surface measure, satisfying $$|K_t(z,w)| \le M t^{-(d-1)} e^{\omega t} \left(1 + \frac{|z - w|}{t}\right)^{-d}$$ for every $t > 0$ and almost every $z, w \in \Gamma$, with $M$ and $\omega$ depending only on the domain and the coefficients?
Source: ter Elst and Ouhabaz (2025), Problem 5, its alternative for real coefficients on Lipschitz 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, Dirichlet-to-Neumann operator, heat kernel bounds, Lipschitz domains
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{terelst2014,
author = {ter Elst, A. F. M. and Ouhabaz, E. M.},
title = {Analysis of the heat kernel of the {D}irichlet-to-{N}eumann operator},
journal = {J. Funct. Anal.},
volume = {267},
number = {11},
year = {2014},
pages = {4066--4109},
doi = {10.1016/j.jfa.2014.09.001},
eprint = {1302.4199},
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 = {58},
doi = {10.1007/s00526-024-02899-y},
eprint = {2404.18272},
archivePrefix = {arXiv},
}
@misc{aim-dtn-poisson-bounds-lipschitz,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Poisson kernel bounds for elliptic boundary diffusion on Lipschitz domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/research/resolved/656-dtn-poisson-bounds-lipschitz.md},
note = {Entry 656 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
No: for d = 2, C = I and the three-quarter disc sector 0 < r < 1, 0 < theta < 3pi/2, no finite M and omega give the bound; the conormal derivative of r^{2/3} sin(2theta/3), of size r^{-1/3} near the reentrant corner, violates the separated-support estimate the bound implies..
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 435-corner-poisson-counterexample/PROOF.md. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00678, "Poisson kernel bounds for elliptic boundary diffusion on Lipschitz domains", clopen.solutions.