Shape optimisation of the first solenoidal Maxwell eigenvalue
Abstract
We study the minimisation of the first solenoidal Maxwell eigenvalue λ1Max(Ω) in perfectly conducting cavities through the scale-invariant functional JPer(Ω):=Per(Ω)λ1Max(Ω) in ℝ3. While the corresponding unconstrained problem is known to be ill-posed, we focus on bounded convex cavities and prove that JPer has a positive infimum in this class. The proof relies on the study of the three main possible behaviours for sequences of convex domains with bounded perimeter: convergence to a bounded convex domain with non-empty interior, collapse to a planar domain or collapse to a one-dimensional domain. The case of a planar collapse is handled through projection arguments, yielding a lower bound, while that of a one-dimensional collapse is controlled by a different argument, showing that in that case JPer must diverge to +∞. Although the question of existence of an optimal shape (that is, ruling out planar collapse) remains open, we derive several negative results and obstructions to existence; typically, no minimiser with a 𝒞3,+ boundary exists. This relies on a fine analysis of first-order optimality conditions.
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BibTeX
@misc{henrot2026shape,
title = {{Shape optimisation of the first solenoidal Maxwell eigenvalue}},
author = {Antoine Henrot and Idriss Mazari and Yannick Privat and Dominik Stantejsky},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21794},
eprint = {2609.21794},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05755062v1},
note = {CC BY-ND 4.0},
}
RIS
TY - UNPB
TI - Shape optimisation of the first solenoidal Maxwell eigenvalue
AU - Antoine Henrot
AU - Idriss Mazari
AU - Yannick Privat
AU - Dominik Stantejsky
PY - 2026
DA - 2026/09/18
PB - HAL
DO - 10.48550/arxiv.2609.21794
UR - https://hal.science/hal-05755062v1
AB - We study the minimisation of the first solenoidal Maxwell eigenvalue λ_1^Max(Ω) in perfectly conducting cavities through the scale-invariant functional J_Per(Ω):=Per(Ω)λ_1^Max(Ω) in ℝ^3. While the corresponding unconstrained problem is known to be ill-posed, we focus on bounded convex cavities and prove that J_Per has a positive infimum in this class. The proof relies on the study of the three main possible behaviours for sequences of convex domains with bounded perimeter: convergence to a bounded convex domain with non-empty interior, collapse to a planar domain or collapse to a one-dimensional domain. The case of a planar collapse is handled through projection arguments, yielding a lower bound, while that of a one-dimensional collapse is controlled by a different argument, showing that in that case J_Per must diverge to +∞. Although the question of existence of an optimal shape (that is, ruling out planar collapse) remains open, we derive several negative results and obstructions to existence; typically, no minimiser with a 𝒞^3,+ boundary exists. This relies on a fine analysis of first-order optimality conditions.
ER -