| Abstract: |
| We consider the Steklov-Wentzell eigenproblem on nearly spherical domains in $\mathbb{R}^{d + 1}$ with $d + 1 \ge 3$, where we include the Laplace-Beltrami operator in the usual Steklov boundary condition. Treating such domains as perturbations of the unit ball, we derive the first-order asymptotic expansion for the Laplace-Beltrami operator and combine this with our previous results from the Steklov eigenproblem to compute the first-order asymptotic expansion for a family of scaled-invariant Steklov-Wentzell eigenvalues. By decomposing the first-order perturbation matrix using the addition theorem for spherical harmonics, we prove that the ball is substationary for an infinite list of scale-invariant Steklov-Wentzell eigenvalues. We extend the analysis to show that the ball is stationary for certain functionals of Steklov-Wentzell eigenvalues and obtain, as a special case, related local isoperimetric results for functionals of Steklov eigenvalues. |
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