Concentric-ball optimality for thermal insulation with radiative heat transfer
Fix $n \ge 2$, $\beta > 0$, $\gamma \ge 0$ and $R > 1$, and put $$\Theta(s) = \beta\Bigl(\frac{s^5}{5} + \gamma s^4 + 2\gamma^2 s^3 + 2\gamma^3 s^2\Bigr).$$ For a bounded Lipschitz open set $\Omega \subset \mathbb{R}^n$ and a measurable set $K \subset \Omega$ let $$E_\Theta(K, \Omega) = \min\Bigl\{\int_\Omega |\nabla v|^2\,dx + \int_{\partial\Omega} \Theta(v)\,d\mathcal{H}^{n-1} : v \in H^1(\Omega),\ 0 \le v \le 1,\ v = 1 \text{ a.e. on } K\Bigr\}.$$ Write $\omega_n = |B_1|$, and let $B_r$ be the ball of radius $r$ with the same centre as $B_1$; for $r = 1$, $E_\Theta(B_1, B_1)$ is the energy of the bare ball $K = \Omega = B_1$, so unused insulation is allowed. Bucur, Nahon, Nitsch and Trombetti studied this energy for a body $K$ held at constant temperature and insulated by the layer $\Omega \setminus K$; up to a constant, $\Theta$ is the boundary law of radiative (Stefan–Boltzmann) heat transfer, $\gamma$ being the ratio of the ambient temperature to the temperature difference. For the convection law $\Theta(s) = \beta s^2$ they proved that, among pairs with $|K| = \omega_n$ and $|\Omega| \le \omega_n R^n$, the minimizers are two concentric balls with outer radius $R$ or $1$, and for every lower semicontinuous nondecreasing $\Theta$ with $\Theta(0) = 0$ they proved that the problem with a volume penalty $\Lambda |\Omega|$ in place of the volume constraint is solved by two concentric balls.
Problem. Is $$E_\Theta(K, \Omega) \ge \min_{1 \le r \le R} E_\Theta(B_1, B_r)$$ for every $n \ge 2$, $\beta > 0$, $\gamma \ge 0$ and $R > 1$, every bounded Lipschitz open set $\Omega \subset \mathbb{R}^n$ and every measurable $K \subset \Omega$ with $|K| = \omega_n$ and $|\Omega| \le \omega_n R^n$?
Source: Bucur, Nahon, Nitsch and Trombetti (2022), the question after Remark 4, whether under reasonable hypotheses on the boundary law their constrained problem is always solved by two concentric balls; Remark 4 names the radiative law among the laws of interest. Stated by AIM — Open Applied Problems for the radiative law, as a concentric-ball inequality.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, radiative heat transfer, shape optimization, thermal insulation
References
@article{bucur2022,
author = {Bucur, Dorin and Nahon, Micka{\"e}l and Nitsch, Carlo and Trombetti, Cristina},
title = {Shape optimization of a thermal insulation problem},
journal = {Calc. Var. Partial Differential Equations},
volume = {61},
number = {5},
year = {2022},
pages = {Paper No. 186},
doi = {10.1007/s00526-022-02298-1},
eprint = {2112.07300},
archivePrefix = {arXiv},
}
@misc{aim-radiative-insulation-concentric-balls,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Concentric-ball optimality for thermal insulation with radiative heat transfer}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/400-radiative-insulation-concentric-balls.md},
note = {Entry 400 at commit 24edaad, 2026-10-15},
}
Resolution claims
We consider a thermal insulation energy with a quadratic Dirichlet term and the nonlinear boundary potential associated with radiative heat transfer. Both the heated set and the surrounding domain may vary, with heated volume normalized to that of the unit ball and an upper bound on the total volume. We prove that the minimum is achieved within the family of …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00825, "Concentric-ball optimality for thermal insulation with radiative heat transfer", clopen.solutions.