Dirichlet-to-Neumann operator on the perturbed unit disk

This article concerns the Laplacian on a perturbed unit disk $Omega_epsilon={ z=rexp(i heta): r <1+epsilon f( heta) }$, with dynamical boundary condition whose solution can be represented by a Dirichlet-to-Neumann semigroup. By neglecting the terms of order $epsilon^2$, we obtain a simple expression which allows us to use Chernoff's theorem for its approximation. As a motivation for this research, we present an example which shows the feasibility of applying Chernoff's Theorem.