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We study the positivity of the second shape derivative around an equilibrium for a functional defined on exterior domains in the plane and which involves the perimeter of the domains and their Dirichlet energy under volume constraint. We prove that small analytic perturbations of circles may be stable or not, depending on the positivity of a simple and explicit two-variable quadratic form. The approach is general and involves a numerical criterion of independent interest for the positivity of a quadratic form on a given hyperplane.

FINITE DIMENSIONAL REDUCTION FOR THE POSITIVITY OF SOME SECOND SHAPE DERIVATIVES