Regularity of optimal shapes for the Dirichlet's energy with volume constraint

Tanguy Briancon

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 1, page 99-122
  • ISSN: 1292-8119

Abstract

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In this paper, we prove some regularity results for the boundary of an open subset of d which minimizes the Dirichlet's energy among all open subsets with prescribed volume. In particular we show that, when the volume constraint is “saturated”, the reduced boundary of the optimal shape (and even the whole boundary in dimension 2) is regular if the state function is nonnegative.

How to cite

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Briancon, Tanguy. "Regularity of optimal shapes for the Dirichlet's energy with volume constraint." ESAIM: Control, Optimisation and Calculus of Variations 10.1 (2010): 99-122. <http://eudml.org/doc/90723>.

@article{Briancon2010,
abstract = { In this paper, we prove some regularity results for the boundary of an open subset of $\xR^d$ which minimizes the Dirichlet's energy among all open subsets with prescribed volume. In particular we show that, when the volume constraint is “saturated”, the reduced boundary of the optimal shape (and even the whole boundary in dimension 2) is regular if the state function is nonnegative. },
author = {Briancon, Tanguy},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Shape optimization; calculus of variations; free boundary; geometrical measure theory.; free boundary; geometrical measure theory},
language = {eng},
month = {3},
number = {1},
pages = {99-122},
publisher = {EDP Sciences},
title = {Regularity of optimal shapes for the Dirichlet's energy with volume constraint},
url = {http://eudml.org/doc/90723},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Briancon, Tanguy
TI - Regularity of optimal shapes for the Dirichlet's energy with volume constraint
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 1
SP - 99
EP - 122
AB - In this paper, we prove some regularity results for the boundary of an open subset of $\xR^d$ which minimizes the Dirichlet's energy among all open subsets with prescribed volume. In particular we show that, when the volume constraint is “saturated”, the reduced boundary of the optimal shape (and even the whole boundary in dimension 2) is regular if the state function is nonnegative.
LA - eng
KW - Shape optimization; calculus of variations; free boundary; geometrical measure theory.; free boundary; geometrical measure theory
UR - http://eudml.org/doc/90723
ER -

References

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  10. B. Gustafsson and H. Shahgholian, Existence and geometric properties of solutions of a free boundary problem in potential theory. J. Reine Angew. Math.473 (1996) 137–179.  Zbl0846.31005
  11. M. Hayouni, Existence et régularité pour des problèmes d'optimisation de formes. Ph.D. thesis, université Henri Poincaré Nancy 1 (1997).  
  12. M. Hayouni, Lipschitz continuity of the state function in a shape optimization problem. J. Convex Anal.6 (1999) 71–90.  Zbl0948.49021
  13. M. Hayouni, T. Briancon and M. Pierre. On a volume constrained shape optimization problem with nonlinear state equation. (to appear).  Zbl1062.49035
  14. X. Pelgrin, Étude d'un problème à frontière libre bidimensionnel. Ph.D. thesis, université Rennes 1 (1994).  
  15. T.H. Wolff, Plane harmonic measures live on sets of σ -finite length. Ark. Mat.31 (1993) 137–172.  Zbl0809.30007

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