Existence and regularity of optimal shapes for spectral functionals with Robin boundary conditions
Mickaël Nahon · arXiv (Cornell University) · 2021
We establish the existence and find some qualitative properties of open sets that minimize functionals of the form $ F(λ_1(Ω;β),\dots,λ_k(Ω;β))$ under measure constraint on $Ω$, where $λ_i(Ω;β)$ designates the $i$-th eigenvalue of the Laplace operator on $Ω$ with Robin boundary conditions of parameter $β>0$. Moreover, we show that minimizers of $λ_k(Ω;β)$ for $k\geq 2$ verify the conjecture $λ_k(Ω;β)=λ_{k-1}(Ω;β)$ in dimension three and more.