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

Abdesslam Boulkhemair; Abdelkrim Chakib; Azeddine Sadik; Abdesslam Boulkhemair; Abdelkrim Chakib; Azeddine Sadik

doi:10.3934/math.20231008

AIMS Mathematics

2023, Volume 8, Issue 8: 19773-19793. doi: 10.3934/math.20231008

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

1.
Laboratoire de Mathématiques Jean Leray, UMR6629 CNRS, UFR Sciences et Techniques, 2 rue de la Houssinière, BP92208, 44322 Nantes, France
2.
Applied Mathematics Team, Faculty of Sciences and Techniques, Sultan Moulay Slimane University, Beni Mellal, Morocco

Received: 24 March 2023 Revised: 17 May 2023 Accepted: 23 May 2023 Published: 13 June 2023
MSC : 35Q93, 46N10, 49Q10, 49Q12

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.
- shape derivative,
- volume functional,
- convex domain,
- star-shaped domain,
- support function,
- gauge function,
- Minkowski deformation,
- shape optimization,
- Brunn-Minkowski theory
Citation: Abdesslam Boulkhemair, Abdelkrim Chakib, Azeddine Sadik. On a shape derivative formula for star-shaped domains using Minkowski deformation[J]. AIMS Mathematics, 2023, 8(8): 19773-19793. doi: 10.3934/math.20231008

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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.

References

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