Journal of Convex Analysis 25 (2018), No. 3, 957--982
Copyright Heldermann Verlag 2018
Topological Derivative: a Semidifferential via the Minkowski Content
Michel C. Delfour
Centre de Recherches Mathématiques, Dép. de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada
delfour@crm.umontreal.ca
The generic notions of shape and topological derivative have proven to be both pertinent and useful from the theoretical and numerical points of view. The shape derivative is a differential while the topological derivative obtained by expansion methods is only a semidifferential. This arises from the fact that the tangent space to the underlying metric spaces of "geometries" is only a cone. We extend its definition to perturbations obtained by creating holes around curves, surfaces, and, potentially, microstructures. In that context, the Hadamard semidifferential that retains the advantages of the standard differential calculus including the chain rule and the fact that semiconvex functions are Hadamard semidifferentiable is a natural notion to study the semidifferentiability of objective functions with respect to the sets/geometries that belong to complete non-linear non-convex metric spaces. An important advantage for state constrained utility functions is that theorems on the one-sided differentiation of minimax of Lagrangians can be used to get the semidifferential. The same adjoint system will occur for shape and topological derivatives. The paper is specialized to the topological derivative on the metric space of characteristic functions.
Keywords: Topological derivative, semidifferential, Minkowski content, Lagrangian, minimax.
MSC: 65K10
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