On a volume constrained variational problem in SBV 2 ( Ω ) : part I

Ana Cristina Barroso; José Matias

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

  • 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 ∈ S B V 2 ( 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 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

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