Analyticity of Steklov Eigenvalues in nearly-hyperspherical domains in \mathbb{R}^{d+1}

TL;DR

This paper proves that Steklov eigenvalues depend analytically on shape perturbations for nearly-hyperspherical domains in dimensions greater than 3, extending previous results from lower dimensions.

Contribution

It establishes the analyticity of the Dirichlet-to-Neumann operator and Steklov eigenvalues under shape perturbations in higher dimensions, generalizing prior lower-dimensional work.

Findings

Steklov eigenvalues are analytic functions of shape perturbation parameters.

The Dirichlet-to-Neumann operator depends analytically on shape perturbations.

Results extend previous work from 2D and 3D to higher dimensions.

Abstract

We consider the Dirichlet-to-Neumann operator (DNO) on nearly-hyperspherical domains in dimension greater than 3. Treating such domains as perturbations of the ball, we prove the analytic dependence of the DNO on the shape perturbation parameter for fixed perturbation functions. Consequently, we conclude that the Steklov eigenvalues are analytic in the shape perturbation parameter as well. To obtain these results, we use the strategy of Nicholls and Nigam (2004), and of Viator and Osting (2020); we transform the Laplace-Dirichlet problem on the perturbed domain to a more complicated, parameter-dependent equation on the ball, and then geometrically bound the Neumann expansion of the transformed DNO. These results are a generalization of the work of Viator and Osting (2020) for dimension 2 and 3.