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).  Zbl0804.28001
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