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

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