| Abstract: |
| We consider the Steklov-Neumann eigenproblem on an open bounded planar domain $\Omega$ with smooth connected boundary $\Gamma$. We pose the extremal eigenvlue problems of minimizing/maximizing the $k$-th non-trivial Steklov-Neumann eigenvalue among boundary partitions into Steklov subdomains $\Gamma_S$ and Neumann subdomains $\Gamma_N$ of prescribed measure. We formulate a relaxation of these EEPs in terms of $L^{\infty}(\Gamma)$ densities rather than partitions of $\Gamma$, and establish existence and optimality conditions for these relaxed EEPs. We also establish a homogenization result that allows solutions of the relaxed EEPs to infer properties of the original EEPs. Given time, we will explore numerical evidence of symmetries of minimizing arrangements of $\Gamma_S$ and $\Gamma_N$ for the $k$-th Steklov-Neumann eigenvalue of the disk. |
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