Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T23:25:44.806Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 February 2004

Tanguy Briancon
Tanguy Briancon*
Affiliation:
Université Rennes 1. Antenne de Bretagne de l'École Normale Supérieure de Cachan; [email protected].
Get access

Abstract

In this paper, we prove someregularity results for the boundary of an open subset of $\xR^d$ whichminimizes the Dirichlet's energy among all open subsets withprescribed volume. In particular we show that, whenthe volume constraint is “saturated”,the reduced boundary of the optimal shape (and even the wholeboundary in dimension 2)is regular if the state function is nonnegative.

Keywords

Shape optimizationcalculus of variationsfree boundarygeometrical measure theory.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 1 , January 2004 , pp. 99 - 122
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

N. Aguilera, H.W. Alt and L.A. Caffarelli, An optimization problem with volume constraint. SIAM J. Control Optimization 24 (1986) 191–198.
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.
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.
T. Briancon, Problèmes de régularité en optimisation de formes. Ph.D. thesis, université Rennes 1 (2002).
M. Crouzeix, Variational approach of a magnetic shaping problem. Eur. J. Mech. 10 (1991) 527–536.
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press (1992).
H. Federer, Geometric measure theory. Springer-Verlag (1969).
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag (1983).
E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser (1986).
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.
M. Hayouni, Existence et régularité pour des problèmes d'optimisation de formes. Ph.D. thesis, université Henri Poincaré Nancy 1 (1997).
M. Hayouni, Lipschitz continuity of the state function in a shape optimization problem. J. Convex Anal. 6 (1999) 71–90.
M. Hayouni, T. Briancon and M. Pierre. On a volume constrained shape optimization problem with nonlinear state equation. (to appear).
X. Pelgrin, Étude d'un problème à frontière libre bidimensionnel. Ph.D. thesis, université Rennes 1 (1994).
T.H. Wolff, Plane harmonic measures live on sets of $\sigma$ -finite length. Ark. Mat. 31 (1993) 137–172.