Best convex shape for insulation of fixed thickness and perimeter
Fix $P, \beta, \delta > 0$. For a compact convex set $K \subset \mathbb{R}^2$ of perimeter $P$, where a line segment is admitted and its perimeter is twice its length, let $\Omega_K = \{x : \operatorname{dist}(x, K) < \delta\}$ and $$I_{\beta,\delta}(K) = \inf \Bigl\{ \int_{\Omega_K} |\nabla v|^2 \, dx + \beta \int_{\partial \Omega_K} v^2 \, d\mathcal{H}^1 : v \in H^1(\Omega_K),\ v \ge 1 \text{ quasi-everywhere on } K \Bigr\},$$ the heat dispersion of the conductor $K$, held at unit temperature and surrounded by an insulating layer of constant thickness $\delta$, with convective (Robin) heat exchange at the outer boundary. Della Pietra, Nitsch and Trombetti proved that among bounded connected planar open sets with piecewise $C^1$ boundary and given perimeter the disk maximizes $I_{\beta,\delta}$, and observed that, by the Blaschke selection theorem and the continuity of $I_{\beta,\delta}$, a minimum over the class above is attained. Bucur, Nahon, Nitsch and Trombetti proved that when the conductor has prescribed volume, the insulated body has volume at most a given bound, and the layer need not have constant thickness, the minimizers are two concentric balls.
Problem. Determine the minimum of $I_{\beta,\delta}$ over this class and characterize all minimizing sets, up to rigid motions, for every triple $(P, \beta, \delta)$.
Source: Della Pietra, Nitsch and Trombetti (2022), §5, Open Problem 1, who ask for the minimum among planar convex sets of given perimeter; the characterization of all minimizers is added by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, convex sets, shape optimization, thermal insulation
References
@article{dellapietra2022,
author = {Della Pietra, Francesco and Nitsch, Carlo and Trombetti, Cristina},
title = {An optimal insulation problem},
journal = {Math. Ann.},
volume = {382},
number = {1-2},
year = {2022},
pages = {745--759},
doi = {10.1007/s00208-020-02058-6},
eprint = {2005.11934},
archivePrefix = {arXiv},
}
@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 = {186},
doi = {10.1007/s00526-022-02298-1},
eprint = {2112.07300},
archivePrefix = {arXiv},
}
@misc{aim-fixed-thickness-insulation-convex-perimeter,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Best convex shape for insulation of fixed thickness and perimeter}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/399-fixed-thickness-insulation-convex-perimeter.md},
note = {Entry 399 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00866, "Best convex shape for insulation of fixed thickness and perimeter", clopen.solutions.