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

Barroso, Ana Cristina; Matias, José

doi:10.1051/cocv:2002009

On a volume constrained variational problem in SBV

^{2} (Ω)

: part I

Barroso, Ana Cristina ; Matias, José

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 223-237

Résumé

We consider the problem of minimizing the energy

E (u) : = \int_{Ω} {| \nabla u (x) |}^{2} d x + \int_{S_{u} \cap Ω} (1 + | [u] (x) |) d H^{N - 1} (x)

among all functions

u \in 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 (\cdot)

is proved and the asymptotic behaviour of the solutions is investigated.

EuDML MR Zbl

DOI : 10.1051/cocv:2002009

Classification : 49J45, 35R35, 76T05
Keywords: special functions of bounded variation, level sets, lower semicontinuity, $\Gamma $-limit

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     author = {Barroso, Ana Cristina and Matias, Jos\'e},
     title = {On a volume constrained variational problem in {SBV}${^2(\Omega )}$ : part {I}},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {223--237},
     year = {2002},
     publisher = {EDP Sciences},
     volume = {7},
     doi = {10.1051/cocv:2002009},
     mrnumber = {1925027},
     zbl = {1047.49016},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2002009/}
}

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Barroso, Ana Cristina; Matias, José. On a volume constrained variational problem in SBV${^2(\Omega )}$ : part I. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 223-237. doi: 10.1051/cocv:2002009

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