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Research Article | Open Access

On a shape derivative formula for star-shaped domains using Minkowski deformation

Abdesslam Boulkhemair1( ), Abdelkrim Chakib2, Azeddine Sadik1,2
Laboratoire de Mathématiques Jean Leray, UMR6629 CNRS, UFR Sciences et Techniques, 2 rue de la Houssinière, BP92208, 44322 Nantes, France
Applied Mathematics Team, Faculty of Sciences and Techniques, Sultan Moulay Slimane University, Beni Mellal, Morocco
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Abstract

We consider the shape derivative formula for a volume cost functional studied in previous papers where we used the Minkowski deformation and support functions in the convex setting. In this work, we extend it to some non-convex domains, namely the star-shaped ones. The formula happens to be also an extension of a well-known one in the geometric Brunn-Minkowski theory of convex bodies. At the end, we illustrate the formula by applying it to some model shape optimization problem.

Keywords

shape derivativevolume functionalconvex domainstar-shaped domainsupport functiongauge functionMinkowski deformationshape optimizationBrunn-Minkowski theory
CLC number: 35Q93, 46N10, 49Q10, 49Q12

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AIMS Mathematics
Volume 8 Issue 8,
August 2023
Pages 19773-19793
DOI: 10.3934/math.20231008

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Cite this article:
Boulkhemair A, Chakib A, Sadik A. On a shape derivative formula for star-shaped domains using Minkowski deformation. AIMS Mathematics, 2023, 8(8): 19773-19793. https://doi.org/10.3934/math.20231008

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Received: 24 March 2023
Revised: 17 May 2023
Accepted: 23 May 2023
Published: 15 August 2023
©2023 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)