Shape and topological derivatives as Hadamard semidifferentials

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Abstract The object of this paper is to further investigate the notion of shape and topological derivatives in the light of the general notion of Hadamard semidifferential for a function defined on a subset of a topological vector space. The use of semitrajectories and the characterization of the adjacent tangent cone provide simple tools for defining Hadamard semi-differentials and differentials without a priori introduction of geometric structures such as, for instance, a differential manifold. Such a simple notion retains all the operations of the classical differential calculus, including the chain rule, for a large class of nondifferentiable functions, in particular, the norms and the convex functions. It also provides a direct access to functions defined on a lousy set or a manifold with boundary. This direct approach is first illustrated in the context of the classical matrix subgroups of the general linear group GL(n) of invertible n×n matrices, which are the prototypes of Lie groups. For the shape derivative we have groups of diffeomorphisms of the Euclidean space ℝ n with the composition operation, and the adjacent tangent cone is a linear space; for the topological derivative we have the group of characteristic functions with the symmetric difference operation and the adjacent tangent cone is only a cone at some points.