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Zbl 0907.49021
Bresch, D.; Simon, J.
On the normal variations of a domain. (Sur les variations normales d'un domaine.) (French)
[J] ESAIM, Control Optim. Calc. Var. 3, 251-261 (1998). ISSN 1292-8119; ISSN 1262-3377/e

Summary: 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 whose 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.

MSC 2000:: *49Q10 Optimization of the shape other than minimal surfaces

Keywords: optimal design; boundary differentiability; domain optimization; normal variations; regularity; transverse variations

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