Abstract
This paper develops boundary integral representation formulas for the second variations of cost functionals for elliptic domain optimization problems. From the collection of all Lipschitz domains Ω which satisfy a constraint ? Ω g(x)?dx=1, a domain is sought which maximizes either $\mathcal{F}_{x_{0}}(\Omega )=F(x_{0},u(x_{0}))$ , fixed x 0?Ω, or ?(Ω)=? Ω F(x,u(x))?dx, where u solves the Dirichlet problem Δu(x)=?f(x), x?Ω, u(x)=0, x??Ω. Necessary and sufficient conditions for local optimality are presented in terms of the first and second variations of the cost functionals $\mathcal{F}_{x_{0}}$ and ?. The second variations are computed with respect to domain variations which preserve the constraint. After first summarizing known facts about the first variations of u and the cost functionals, a series of formulas relating various second variations of these quantities are derived. Calculating the second variations depends on finding first variations of solutions u when the data f are permitted to depend on the domain Ω.
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