Hadamard semidifferential, oriented distance function, and some applications

Michel C. Delfour

doi:10.3934/cpaa.2021076

Article Contents

2022, Volume 21, Issue 6: 1917-1951. Doi: 10.3934/cpaa.2021076 open access

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Hadamard semidifferential, oriented distance function, and some applications

Michel C. Delfour

Département de mathématiques et de statistique and Centre de recherches mathématiques, Université de Montréal, CP 6128, succ. Centre-ville, Montréal (Qc), Canada H3C 3J7

Received: December 2020

Revised: March 2021

Early access: April 2021

Published: June 2022

This research was supported by the Natural Sciences and Engineering research Council of Canada through Discovery Grants RGPIN-05279-2017 and a Grant from the Collaborative research and Training Experience (CREATE) program in Simulation-based Engineering Science.

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Abstract

The Hadamard semidifferential calculus preserves all the operations of the classical differential calculus including the chain rule for a large family of non-differentiable functions including the continuous convex functions. It naturally extends from the $ n $-dimensional Euclidean space $ \operatorname{\mathbb R}^n $ to subsets of topological vector spaces. This includes most function spaces used in Optimization and the Calculus of Variations, the metric groups used in Shape and Topological Optimization, and functions defined on submanifolds.

Certain set-parametrized functions such as the characteristic function $ \chi_A $of a set $ A $, the distance function $ d_A $ to $ A $, and the oriented (signed) distance function $ b_A = d_A-d_{ \operatorname{\mathbb R}^n\backslash A} $ can be used to identify a space of subsets of $ \operatorname{\mathbb R}^n $ with a metric space of set-parametrized functions. Many geometrical properties of domains (convexity, outward unit normal, curvatures, tangent space, smoothness of boundaries) can be expressed in terms of the analytical properties of $ b_A $ and a simple intrinsic differential calculus is available for functions defined on hypersurfaces without appealing to local bases or Christoffel symbols.

The object of this paper is to extend the use of the Hadamard semidifferential and of the oriented distance function from finite to infinite dimensional spaces with some selected illustrative applications from shapes and geometries, plasma physics, and optimization.

Keywords:

Differentiation theory,
differentiable maps,
calculus of functions between topological vector spaces,
optimization of shapes,
sensitivity analysis,
calculus of variations,
plasma physics.

Mathematics Subject Classification: Primary: 58C20, 58C25, 46G05, 46T20, 26E15, 26E20, 49Q10, 49Q12.

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