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

top
In this paper, we prove some regularity results for the boundary of an open subset of 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

top

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

top
  1. N. Aguilera, H.W. Alt and L.A. Caffarelli, An optimization problem with volume constraint. SIAM J. Control Optimization24 (1986) 191–198.  
  2. H.W. Alt and L.A. Caffarelli, Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math.325 (1981) 105–144.  
  3. H.W. Alt, L.A. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc.282 (1984) 431–461.  
  4. T. Briancon, Problèmes de régularité en optimisation de formes. Ph.D. thesis, université Rennes 1 (2002).  
  5. M. Crouzeix, Variational approach of a magnetic shaping problem. Eur. J. Mech.10 (1991) 527–536.  
  6. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press (1992).  
  7. H. Federer, Geometric measure theory. Springer-Verlag (1969).  
  8. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag (1983).  
  9. E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser (1986).  
  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.  
  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.  
  13. M. Hayouni, T. Briancon and M. Pierre. On a volume constrained shape optimization problem with nonlinear state equation. (to appear).  
  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 -finite length. Ark. Mat.31 (1993) 137–172.  

NotesEmbed ?

top

You must be logged in to post comments.