A REMARK ON THE LOSS OF REGULARITY DUE TO NORMAL VARIATIONS OF A DOMAIN

In domain optimization and in free boundary problems, normal variation s of a reference domain are frequently used. We prove that such variat ions do not preserve the regularity of the domain. More precisely, me give a domain Omega whose boundary Gamma is of class C-m and a variati on alpha of class C-infinity such that Gamma + alpha n is not of class C-m. Moreover, for any epsilon > 0, the boundary Gamma + alpha n is n ot of class C-m-1+epsilon. The variation. alpha may be chosen as small as required, and m may be integer or not. Therefore, the use of norma l variations in an iterative approximation method for domain optimizat ion brings a loss of regularity at each iteration.