Steklov eigenvalues of nearly circular area-normalized domains

Abstract: We consider Steklov eigenvalues of nearly circular domains in $\R^{2}$ of fixed unitary area. In \cite{viator2018}, the authors treated such domains as perturbations of the disk, and they computed the first-order term of the asymptotic expansions of the Steklov eigenvalues for reflection-symmetric perturbations; here, we expand these first-order results beyond reflection-symmetry. We also recover the second-order asymptotic expansions, which enable us to prove that no Steklov eigenvalue beyond the first positive one is locally shape-optimized by the disk.