We consider the minimization of averaged shape optimization problems over the class of sets of finite perimeter. We use occupational measures, which are probability measures defined in terms of the reduced boundary of sets of finite perimeter, that allow to transform the minimization into a linear problem on a set of measures. The averaged nature of the problem allows the optimal value to be approximated with sets with unbounded perimeter. In this case, we show that we can also approximate the optimal value with convex polytopes with n+1 faces shrinking to a point. We derive conditions under which we show the existence of minimizers and we also analyze the appropriate spaces in which to study the problem.
Accepted:
DOI: 10.1051/cocv/2017017
Keywords: Shape optimization, occupational measures, sets of finite perimeter, Cheeger sets
@article{COCV_2018__24_3_1141_0, author = {Bright, Ido and Li, Qinfeng and Torres, Monica}, title = {Occupational measures and averaged shape optimization}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1141--1165}, publisher = {EDP-Sciences}, volume = {24}, number = {3}, year = {2018}, doi = {10.1051/cocv/2017017}, zbl = {1405.49030}, mrnumber = {3877196}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017017/} }
TY - JOUR AU - Bright, Ido AU - Li, Qinfeng AU - Torres, Monica TI - Occupational measures and averaged shape optimization JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1141 EP - 1165 VL - 24 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017017/ DO - 10.1051/cocv/2017017 LA - en ID - COCV_2018__24_3_1141_0 ER -
%0 Journal Article %A Bright, Ido %A Li, Qinfeng %A Torres, Monica %T Occupational measures and averaged shape optimization %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1141-1165 %V 24 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017017/ %R 10.1051/cocv/2017017 %G en %F COCV_2018__24_3_1141_0
Bright, Ido; Li, Qinfeng; Torres, Monica. Occupational measures and averaged shape optimization. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 3, pp. 1141-1165. doi : 10.1051/cocv/2017017. http://archive.numdam.org/articles/10.1051/cocv/2017017/
[1] Monographs in Mathematics. Translation ofthe 1950 Russian, edited by , and Springer Verlag, Berlin (2005) | MR | Zbl
,[2] Uniqueness of the Cheeger set of a convex body. Nonlin. Anal. 70 (2009) 32–44 | DOI | MR | Zbl
and ,[3] A characterization of Convex Calibrable Sets in ℝ^{n}. Math. Ann. 332 (2005) 329–366 | DOI | MR | Zbl
, and ,[4] Connected components of sets of finite perimeter and applications to image processing. J. Europ. Math. Soc. 3 (2001) 39–92 | DOI | MR | Zbl
, , and ,[5] Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press: New York (2000) | MR | Zbl
, and ,[6] Periodic optimization suffices for infinite horizon planar optimal control. SIAM J. Control Optim. 48 (2010) 4963–4986 | DOI | MR | Zbl
and ,[7] A note on flux integrals over smooth regular domains. Pacific J. Math. 272 (2014) 305–322 | DOI | MR | Zbl
and ,[8] The integral of the normal and fluxes over sets of finite perimeter. Interface Free Bound. 17 (2015) 245–259 | DOI | MR | Zbl
and ,[9] Convergence of probability measures. Wiley Interscience (2009) | Zbl
,[10] On the selection of maximal Cheeger sets. Differ. Integral Equ. 20 (2007) 991–1004. | MR | Zbl
, and ,[11] Fine properties of the subdifferential for a class of one-homogeneous functionals. (2012). | arXiv | Zbl
, and ,[12] Uniqueness of the Cheeger set of a convex body. Pacific J. Math. 231 (2007) 77–90 | DOI | MR | Zbl
, and ,[13] Some Remarks on uniqueness and regularity of Cheeger sets. Rend. Sem. Mat. Univ. Padova 123 (2010) | DOI | Numdam | MR | Zbl
, and ,[14] Approximation of maximal Cheeger sets by projection. Math. Model. Number Anal. 43 (2008) 139–150 | DOI | Numdam | MR | Zbl
, and ,[15] A lower bound for the smallest eigenvalue of the Laplacian. Problems in analysis, Princeton Univ. Press. New Jersey (1970) 195–199. | MR | Zbl
,[16] Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147 (1999) 89–118. | DOI | MR | Zbl
and ,[17] Divergence-measure fields, sets of finite perimeter, and conservation laws. Arch. Rational Mech. Anal. 175 (2005) 245–267 | DOI | MR | Zbl
and ,[18] Gauss–Green Theorem for Weakly Differentiable Vector Fields, Sets of Finite Perimeter, and Balance Laws. Commun. Pure Appl. Math. 62 (2009) 242–304 | DOI | MR | Zbl
, and ,[19] Hyperbolic Conservation Laws in Continuum Physics, 2nd edition. Springer Verlag: Berlin (2005) | DOI | MR | Zbl
,[20] Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton, FL (1992). | MR | Zbl
and ,[21] Geometric Measure Theory. Springer Verlag, New York Inc. (1969). | MR | Zbl
,[22] A note on Cheeger sets. Proc. of Amer. Math. Soc. 137 (2009) 2057–2062 | DOI | MR | Zbl
, and ,[23] Duality in linear programming problems related to deterministic long run average problems of optimal control. SIAM J. Control Optim. 47 (2008) 1667–1700 | DOI | MR | Zbl
, and ,[24] Linear programming approach to deterministic infinite horizon optimal control problems with discounting. SIAM J. Control Optim. 48 (2009) 2480–2512 | DOI | MR | Zbl
and ,[25] Convex and discrete geometry. Springer, Berlin (2007) | MR | Zbl
,[26] Generalized Cheeger sets related to landslides. Calcul. Variat. Partial Differ. Equ. 23 (2005) 227–249 | DOI | MR | Zbl
and ,[27] Sets of Finite Perimeter and Geometric Variational Problems. Cambridge (2011) | MR | Zbl
,[28] Convex Analysis. Princeton, New Jersey, Princeton University Press (1970) | DOI | MR | Zbl
,[29] The Mechanics and Thermodynamics of Continuous Media. Springer Verlag (1997) | DOI | MR | Zbl
[30] Divergence measure fields and Cauchy’s stress theorem. Rend. Sem. Mat. Univ. Padova 113 (2005) 15–45 | Numdam | MR | Zbl
[31] Weakly Differentiable Functions. In Vol. 120 of Graduate Texts in Mathematics. Springer Verlag: New York (1989) | DOI | MR | Zbl
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