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On a Volume Constrained Variational Problem in SBV²(Ω): Part I

Ana Cristina Barroso; José Matias

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

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

Abstract

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We consider the problem of minimizing the energy E ( u ) : = ∫ Ω | ∇ u ( x ) | 2 d x + ∫ S u ∩ Ω 1 + | [ u ] ( x ) | d H N - 1 ( x ) among all functions u ∈ SBV²(Ω) for which two level sets { u = l i } have prescribed Lebesgue measure α i . Subject to this volume constraint the existence of minimizers for E(.) is proved and the asymptotic behaviour of the solutions is investigated.

How to cite

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Barroso, Ana Cristina, and Matias, José. "On a Volume Constrained Variational Problem in SBV²(Ω): Part I." ESAIM: Control, Optimisation and Calculus of Variations 7 (2010): 223-237. <http://eudml.org/doc/90619>.

@article{Barroso2010,
abstract = { We consider the problem of minimizing the energy $$ E(u):= \int\_\{\Omega\}|\nabla u(x)|^2 \, \{\rm d\}x + \int\_\{S\_u \cap \Omega\}\left (1 + |[u](x)|\right) \, \{\rm d\}H^\{N - 1\}(x)$$ among all functions u ∈ SBV²(Ω) for which two level sets $\\{u = l_i\\}$ have prescribed Lebesgue measure $\alpha_i$. Subject to this volume constraint the existence of minimizers for E(.) 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; Γ-limit.; special functions of bounded variation; lower semicontinuity; -limit; SBV},
language = {eng},
month = {3},
pages = {223-237},
publisher = {EDP Sciences},
title = {On a Volume Constrained Variational Problem in SBV²(Ω): Part I},
url = {http://eudml.org/doc/90619},
volume = {7},
year = {2010},
}

TY - JOUR
AU - Barroso, Ana Cristina
AU - Matias, José
TI - On a Volume Constrained Variational Problem in SBV²(Ω): Part I
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 7
SP - 223
EP - 237
AB - We consider the problem of minimizing the energy $$ E(u):= \int_{\Omega}|\nabla u(x)|^2 \, {\rm d}x + \int_{S_u \cap \Omega}\left (1 + |[u](x)|\right) \, {\rm d}H^{N - 1}(x)$$ among all functions u ∈ SBV²(Ω) for which two level sets $\{u = l_i\}$ have prescribed Lebesgue measure $\alpha_i$. Subject to this volume constraint the existence of minimizers for E(.) is proved and the asymptotic behaviour of the solutions is investigated.
LA - eng
KW - Special functions of bounded variation; level sets; lower semicontinuity; Γ-limit.; special functions of bounded variation; lower semicontinuity; -limit; SBV
UR - http://eudml.org/doc/90619
ER -

References

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  8. G. Dal Maso, An Introduction to Γ-convergence. Birkhäuser (1993).  
  9. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Stud. Adv. Math. (1992).  
  10. E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Birkhäuser (1984).  
  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.  
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