# On the normal variations of a domain

• Volume: 3, page 251-261
• ISSN: 1292-8119

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## Abstract

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In domain optimization problems, normal variations of a reference domain are frequently used. We prove that such variations do not preserve the regularity of the domain. More precisely, we give a bounded domain which boundary is m times differentiable and a scalar variation which is infinitely differentiable such that the deformed boundary is only m-1 times differentiable. We prove in addition that the only normal variations which preserve the regularity are those with constant magnitude. This shows that the use of normal variations in an iterative approximation method for domain optimization generates a loss of regularity at each iteration, and thus it is better to use transverse variations which preserve the regularity of the domain.

## How to cite

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Bresch, D., and Simon, J.. "On the normal variations of a domain ." ESAIM: Control, Optimisation and Calculus of Variations 3 (2010): 251-261. <http://eudml.org/doc/197339>.

@article{Bresch2010,
abstract = { In domain optimization problems, normal variations of a reference domain are frequently used. We prove that such variations do not preserve the regularity of the domain. More precisely, we give a bounded domain which boundary is m times differentiable and a scalar variation which is infinitely differentiable such that the deformed boundary is only m-1 times differentiable. We prove in addition that the only normal variations which preserve the regularity are those with constant magnitude. This shows that the use of normal variations in an iterative approximation method for domain optimization generates a loss of regularity at each iteration, and thus it is better to use transverse variations which preserve the regularity of the domain. },
author = {Bresch, D., Simon, J.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Domain variations; optimal design; boundary differentiability. ; boundary differentiability; domain optimization; normal variations; regularity; transverse variations},
language = {eng},
month = {3},
pages = {251-261},
publisher = {EDP Sciences},
title = {On the normal variations of a domain },
url = {http://eudml.org/doc/197339},
volume = {3},
year = {2010},
}

TY - JOUR
AU - Bresch, D.
AU - Simon, J.
TI - On the normal variations of a domain
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 3
SP - 251
EP - 261
AB - In domain optimization problems, normal variations of a reference domain are frequently used. We prove that such variations do not preserve the regularity of the domain. More precisely, we give a bounded domain which boundary is m times differentiable and a scalar variation which is infinitely differentiable such that the deformed boundary is only m-1 times differentiable. We prove in addition that the only normal variations which preserve the regularity are those with constant magnitude. This shows that the use of normal variations in an iterative approximation method for domain optimization generates a loss of regularity at each iteration, and thus it is better to use transverse variations which preserve the regularity of the domain.
LA - eng
KW - Domain variations; optimal design; boundary differentiability. ; boundary differentiability; domain optimization; normal variations; regularity; transverse variations
UR - http://eudml.org/doc/197339
ER -

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