Some properties of the eigenvalues of the volume Wentzell-Laplace operator
Résumé
In this paper, we consider the volume Wentzell-Laplace eigenvalues problem. In order to illustrate the behavior of the spectrum, we consider an explicit example about the ball in R^2 , where we find explicitly the eigenvalues and their corresponding eigenfunctions. After showing the analyticity of the eigenvalues λ and their corresponding eigenfunctions u in an open neighborhood of t = 0, we derive, in the sense of Hadamard, the first and second-order shape derivatives of the eigenvalues at time t = 0. Moreover, after deriving two new Pohožaev's identities, we show that the ball is not a critical shape of this eigenvalues problem, with and without a volume constraint, in any dimension.
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