# On a volume constrained variational problem in SBV${}^{2}\left(\Omega \right)$ : part I

Ana Cristina Barroso; José Matias

ESAIM: Control, Optimisation and Calculus of Variations (2002)

- Volume: 7, page 223-237
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topBarroso, Ana Cristina, and Matias, José. "On a volume constrained variational problem in SBV${^2(\Omega )}$ : part I." ESAIM: Control, Optimisation and Calculus of Variations 7 (2002): 223-237. <http://eudml.org/doc/245983>.

@article{Barroso2002,

abstract = {We consider the problem of minimizing the energy\[\hspace*\{-42.67912pt\}E(u):= \int \_\{\Omega \}|\nabla u(x)|^2 \, \{\rm d\}x + \int \_\{S\_u \cap \Omega \}\left(1 + |[u](x)|\right) \, \{\rm d\}H^\{N - 1\}(x)\]among all functions $u \in SBV^2(\Omega )$ for which two level sets $\lbrace u = l_i\rbrace $ have prescribed Lebesgue measure $\alpha _i$. Subject to this volume constraint the existence of minimizers for $E(\cdot )$ is proved and the asymptotic behaviour of the solutions is investigated.},

author = {Barroso, Ana Cristina, Matias, José},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {special functions of bounded variation; level sets; lower semicontinuity; $\Gamma $-limit; -limit; SBV},

language = {eng},

pages = {223-237},

publisher = {EDP-Sciences},

title = {On a volume constrained variational problem in SBV$\{^2(\Omega )\}$ : part I},

url = {http://eudml.org/doc/245983},

volume = {7},

year = {2002},

}

TY - JOUR

AU - Barroso, Ana Cristina

AU - Matias, José

TI - On a volume constrained variational problem in SBV${^2(\Omega )}$ : part I

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2002

PB - EDP-Sciences

VL - 7

SP - 223

EP - 237

AB - We consider the problem of minimizing the energy\[\hspace*{-42.67912pt}E(u):= \int _{\Omega }|\nabla u(x)|^2 \, {\rm d}x + \int _{S_u \cap \Omega }\left(1 + |[u](x)|\right) \, {\rm d}H^{N - 1}(x)\]among all functions $u \in SBV^2(\Omega )$ for which two level sets $\lbrace u = l_i\rbrace $ have prescribed Lebesgue measure $\alpha _i$. Subject to this volume constraint the existence of minimizers for $E(\cdot )$ is proved and the asymptotic behaviour of the solutions is investigated.

LA - eng

KW - special functions of bounded variation; level sets; lower semicontinuity; $\Gamma $-limit; -limit; SBV

UR - http://eudml.org/doc/245983

ER -

## References

top- [1] L. Ambrosio, A compactness theorem for a special class of functions of bounded variation. Boll. Un. Mat. Ital. 3-B (1989) 857-881. Zbl0767.49001MR1032614
- [2] L. Ambrosio, I. Fonseca, P. Marcellini and L. Tartar, On a volume constrained variational problem. Arch. Rat. Mech. Anal. 149 (1999) 23-47. Zbl0945.49005MR1723033
- [3] N. Aguilera, H.W. Alt and L.A. Caffarelli, An optimization problem with volume constraint. SIAM J. Control Optim. 24 (1986) 191-198. Zbl0588.49005MR826512
- [4] 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. Zbl0449.35105MR618549
- [5] A. Braides and V. Chiadò–Piat, Integral representation results for functionals defined on $SBV(\Omega ;{\mathbb{R}}^{m})$. J. Math. Pures Appl. 75 (1996) 595-626. Zbl0880.49010
- [6] G. Congedo and L. Tamanini, On the existence of solutions to a problem in multidimensional segmentation. Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1991) 175-195. Zbl0729.49003MR1096603
- [7] E. De Giorgi and L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni. Atti Accad. Naz. Lincei 82 (1988) 199-210. Zbl0715.49014MR1152641
- [8] G. Dal Maso, An Introduction to $\Gamma $-convergence. Birkhäuser (1993). Zbl0816.49001MR1201152
- [9] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Stud. Adv. Math. (1992). Zbl0804.28001MR1158660
- [10] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser (1984). Zbl0545.49018MR775682
- [11] M.E. Gurtin, D. Polignone and J. Vinals, Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6 (1996) 815-831. Zbl0857.76008MR1404829
- [12] P. Tilli, On a constrained variational problem with an arbitrary number of free boundaries. Interf. Free Boundaries 2 (2000) 201-212. Zbl0995.49002MR1760412
- [13] W. Ziemer, Weakly Differentiable Functions. Springer-Verlag (1989). Zbl0692.46022MR1014685

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.