The Green function of the Laplacian with the homogeneous Dirichlet boundary condition on bounded domains is considered. The variation of the Green function with respect to domain perturbations is called the Hadamard variation. In this paper, we present a unified approach to deriving the Hadamard variation. In our approach, the classical first Hadamard variation is obtained in a natural way under a less restrictive regularity assumption on the boundary smoothness. Furthermore, the second Hadamard variational formula with respective to general domain perturbations is obtained, which is an extension of the classical result of Garabedian–Schiffer in which only normal perturbation is considered.
Takashi SUZUKI. Takuya TSUCHIYA. "First and second Hadamard variational formulae of the Green function for general domain perturbations." J. Math. Soc. Japan 68 (4) 1389 - 1419, October, 2016. https://doi.org/10.2969/jmsj/06841389