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Minimax Lagrangian Approach to the Differentiability of Nonlinear PDE Constrained Shape Functions Without Saddle Point Assumption

The object of this paper is the computation of the domain or boundary expression of a state constrained shape function without explicitly resorting to the material derivative. Our main theorem bypasses the restrictive saddle point assumption in existing differentiability theorems of minimax by introducing the notion of the averaged adjoint state, thus extending the use of minimax theorems to some classes of nonlinear state equations. As an illustration, the theorem is applied to a shape function that depends on a quasi-linear transmission problem. Using a Gagliardo penalization the existence of optimal shapes is established.