Hadamard Differential Calculus and Applications

The Hadamard differential was introduced in 1923 by Hadamard and promoted in 1937 by Fréchet who extended it to vector spaces of functions. Infinite dimension is equivalent to the Fréchet differential introduced in 1911, but in function spaces, Hadamard is more general than Fréchet, which is restricted to normed vector spaces. In 1978 Penot gave the appropriate definition of a semi-differential in the sense of Hadamard by using semi-paths with a semi-tangent.

The Hadamard semidifferentiable functions are probably the largest family of nondifferentiable functions that retain all the features of the classical differential calculus including the chain rule. Continuous convex and semiconvex functions and all norms are Hadamard semidifferentiable. They have a very large intersection with the family of functions studied in Nonsmooth Analysis (Clarke), but they are not contained into one another. Because of its geometric character, the notion of Hadamard semidifferential readily extends to functions defined on smooth embedded differential submanifolds. It also naturally extends to metric groups that naturally occur in optimization problems with respect to the geometry to make sense of the so-called shape and topological derivatives. Applications to Danskin's Theorem and a problem in Plasma Physics will be briefly described.