×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] L. Ambrosio - G. Buttazzo , An optimal design problem with perimeter penalization , Calc. Var. 1 ( 1993 ), 55 - 69 . MR 1261717 | Zbl 0794.49040 · Zbl 0794.49040
[2] M. Berger , Géométrie , vol. 3 , Cedic/Fernand Nathan , Paris , 1978 . MR 536872 | Zbl 0423.51003 · Zbl 0423.51003
[3] M. Berger - B. Gostiaux , Géométrie Différentielle , Armand Collin , Paris , 1987 . MR 494180 · Zbl 0619.53001
[4] D. Bucur - J.P. Zolésio , Continuité par rapport au domaine dans le problème de Neumann , C.R. Acad. Sci. Paris Sér. I Math. 319 ( 1994 ), 57 - 60 . MR 1285898 | Zbl 0805.35010 · Zbl 0805.35010
[5] D. Bucur - J.P. Zolésio , Free Boundary Problems and Density Perimeter , J. Differential Equations 126 ( 1996 ), 224 - 243 . MR 1383977 | Zbl 0856.35137 · Zbl 0856.35137
[6] M. Delfour - J.P. Zolésio , Shape Optimization: Oriented Distance Function, Comett. Cours, Sophia Antipolis , 1993 .
[7] H. Federer , Geometric Measure Theory , Springer Verlag , Berlin , 1969 . MR 257325 | Zbl 0176.00801 · Zbl 0176.00801
[8] S.L. Kulkarni - S. Mitter - T. Richardson , An Existence Theorem and Lattice Approximations for a Variational Problem Arising in Computer Vision, Signal Processing, Part I Signal Processing Theory , L. Auslander, T. Kailath and S. Mitter eds., IMA series, Springer-Verlag , 1990 , p. 189 - 210 . MR 1044605 | Zbl 0701.49003 · Zbl 0701.49003
[9] J. Sokolowski - J.P. Zolésio , Introduction to Shape Optimization , Springer Verlag , 1992 . MR 1215733 | Zbl 0761.73003 · Zbl 0761.73003
[10] W. Ziemer , Weakly Differentiable Functions , Springer , 1989 . MR 1014685 | Zbl 0692.46022 · Zbl 0692.46022
[11] J.P. Zolésio , Weak Shape Formulation of Free Boundary Problems , Ann. Scuola Norm. Sup. Pisa Cl. Sci. 21 ( 1994 ), 11 - 44 . Numdam | MR 1276761 | Zbl 0807.49018 · Zbl 0807.49018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.