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Boundary optimization under pseudo curvature constraint. (English) Zbl 0889.49026
The paper is devoted to the existence theory for shape optimization problems, assuming that the cost functional contains a term related to the perimeter of the unknown set. In particular, the authors introduce the $$(\gamma,H)$$-density perimeter $P_{\gamma,H}(A):=\sup_{0<\varepsilon<\gamma} \Biggl\{ {\text{meas}(A_\varepsilon)\over 2\varepsilon}+H(\varepsilon)\Biggr\}$ (with $$\gamma>0$$ and $$H:[0,\infty)\to{\mathbb{R}}$$ continuous) and compare this concept with more classical notions of perimeter, due to De Giorgi, Minkowski, Hausdorff. Moreover, they show that the Hausdorff convergence topology and the $$L^1$$ topology on sets are comparable under uniform bounds on the $$(\gamma,H)$$-perimeter. A related concept of density curvature is introduced, and it is proved that the $$(\gamma,H)$$-density perimeter $$\Gamma$$-converges as $$\gamma\downarrow 0$$ to the Minkowki content in the class of sets with uniformly bounded density curvature. Applications to several shape optimization problems are presented in the last part of the paper.
Reviewer: L.Ambrosio (Pavia)

##### MSC:
 49Q10 Optimization of shapes other than minimal surfaces
##### Keywords:
shape optimization; perimeter; Minkowski content
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##### References:
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