# Analyticity of Steklov eigenvalues of nearly-circular and nearly-spherical domains

## TL;DR

This paper proves that Steklov eigenvalues depend analytically on small domain perturbations for nearly-circular and nearly-spherical shapes, using a transformation approach and geometric bounds.

## Contribution

It establishes the analyticity of the Dirichlet-to-Neumann operator and Steklov eigenvalues under domain perturbations, extending previous methods to nearly-circular and nearly-spherical domains.

## Key findings

- Steklov eigenvalues are analytic functions of domain perturbation parameters.
- The Dirichlet-to-Neumann operator's analyticity is proven for nearly-circular and nearly-spherical domains.
- Transformation and geometric bounding techniques are effectively used for the analysis.

## Abstract

We consider the Dirichlet-to-Neumann operator (DNO) on nearly-circular and nearly-spherical domains in two and three dimensions, respectively. Treating such domains as perturbations of the ball, we prove the analyticity of the DNO with respect to the domain perturbation parameter. Consequently, the Steklov eigenvalues are also shown to be analytic in the domain perturbation parameter. To obtain these results, we use the strategy of Nicholls and Nigam (2004); we transform the equation on the perturbed domain to a ball and geometrically bound the Neumann expansion of the transformed Dirichlet-to-Neumann operator.