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On a volume constrained variational problem in \(SBV{^2(\Omega)}\). I. (English) Zbl 1047.49016

Authors’ abstract: “We consider the problem of minimizing the energy \(E(u):=\int _{\Omega}| \nabla u(x)| ^{2}\,dx+\int_{S_{u}\cap\Omega}(1+| [u](x)| )\,dH^{n-1}(x)\) among all functions \(u\in SBV^{2}(\Omega)\) for which two level sets \(\{u=l_{i}\}\) have prescribed Lebesgue measure \(\alpha_{i}.\) Subject to this volume constraint the existence of minimizers for \(E(\cdot)\) is proved and the asymptotic behaviour of the solutions is investigated.”

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35R35 Free boundary problems for PDEs
35A15 Variational methods applied to PDEs
49K10 Optimality conditions for free problems in two or more independent variables
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References:

[1] L. Ambrosio , A compactness theorem for a special class of functions of bounded variation . Boll. Un. Mat. Ital. 3-B ( 1989 ) 857 - 881 . MR 1032614 | Zbl 0767.49001 · Zbl 0767.49001
[2] L. Ambrosio , I. Fonseca , P. Marcellini and L. Tartar , On a volume constrained variational problem . Arch. Rat. Mech. Anal. 149 ( 1999 ) 23 - 47 . MR 1723033 | Zbl 0945.49005 · Zbl 0945.49005 · doi:10.1007/s002050050166
[3] N. Aguilera , H.W. Alt and L.A. Caffarelli , An optimization problem with volume constraint . SIAM J. Control Optim. 24 ( 1986 ) 191 - 198 . MR 826512 | Zbl 0588.49005 · Zbl 0588.49005 · doi:10.1137/0324011
[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 . MR 618549 | Zbl 0449.35105 · Zbl 0449.35105
[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 . Zbl 0880.49010 · Zbl 0880.49010
[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 . Numdam | MR 1096603 | Zbl 0729.49003 · Zbl 0729.49003
[7] E. De Giorgi and L. Ambrosio , Un nuovo tipo di funzionale del calcolo delle variazioni . Atti Accad. Naz. Lincei 82 ( 1988 ) 199 - 210 . MR 1152641 | Zbl 0715.49014 · Zbl 0715.49014
[8] G. Dal Maso , An Introduction to \(\Gamma \)-convergence . Birkhäuser ( 1993 ). MR 1201152 | Zbl 0816.49001 · Zbl 0816.49001
[9] L.C. Evans and R.F. Gariepy , Measure Theory and Fine Properties of Functions . CRC Press, Stud. Adv. Math. ( 1992 ). MR 1158660 | Zbl 0804.28001 · Zbl 0804.28001
[10] E. Giusti , Minimal Surfaces and Functions of Bounded Variation . Birkhäuser ( 1984 ). MR 775682 | Zbl 0545.49018 · Zbl 0545.49018
[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 . MR 1404829 | Zbl 0857.76008 · Zbl 0857.76008 · doi:10.1142/S0218202596000341
[12] P. Tilli , On a constrained variational problem with an arbitrary number of free boundaries . Interf. Free Boundaries 2 ( 2000 ) 201 - 212 . MR 1760412 | Zbl 0995.49002 · Zbl 0995.49002 · doi:10.4171/IFB/18
[13] W. Ziemer , Weakly Differentiable Functions . Springer-Verlag ( 1989 ). MR 1014685 | Zbl 0692.46022 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
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