Leonardi, Gian Paolo:
$\Gamma$-convergence of constrained Dirichlet functionals
Bollettino dell'Unione Matematica Italiana Serie 8 6-B (2003), fasc. n.2, p. 339-351, Unione Matematica Italiana (English)
pdf (277 Kb), djvu (187 Kb). | MR1988209 | Zbl 1177.49026
Sunto
Dato $\Omega\subset \mathbb{R}^{n}$ aperto, limitato e connesso, con frontiera Lipschitziana e volume $|\Omega|$, si prova che la successione $\mathcal{F}_{k}$ di funzionali di Dirichlet definiti in $H^{1}(\Omega; \mathbb{R}^{d})$, con vincoli di volume $v^{k}$ su $m\geq2$ insiemi di livello prescritti, tali che $\sum_{i=1}^{m}v_{i}^{k}< |\Omega|$ per ogni $k$, $\Gamma$-converge, quando $v^{k}\rightarrow v$ con $\sum_{i=1}^{m}v_{i}^{k}=|\Omega|$, al quadrato della variazione totale in $BV(V; \mathbb{R}^{d})$, con vincoli di volume $v$ sui medesimi insiemi di livello.
Referenze Bibliografiche
[1]
H. W. ALT-
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[2]
L. AMBROSIO,
Corso introduttivo alla Teoria Geometrica della Misura ed alle superfici minime,
Scuola Norm. Sup., Pisa,
1997. |
MR 1736268 |
Zbl 0977.49028[3]
L. AMBROSIO-
I. FONSECA-
P. MARCELLINI-
L. TARTAR,
On a volume-constrained variational problem,
Arch. Ration. Mech. Anal.,
149 (
1999), 23-47. |
MR 1723033 |
Zbl 0945.49005[4]
L. AMBROSIO-
N. FUSCO-
D. PALLARA,
Functions of bounded variation and free discontinuity problems,
The Clarendon Press Oxford University Press, New York,
2000. Oxford Science Publications. |
MR 1857292 |
Zbl 0957.49001[5]
S. BALDO,
Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids,
Ann. Inst. H. Poincare Anal. Non Lineaire,
7 (
1990), 67-90. |
fulltext mini-dml |
MR 1051228 |
Zbl 0702.49009[6]
G. DAL MASO,
An introduction to $\Gamma$-convergence,
Birkhauser Boston Inc., Boston, MA,
1993. |
MR 1201152 |
Zbl 0816.49001[7]
L. C. EVANS-
R. F. GARIEPY,
Lecture Notes on Measure Theory and Fine Properties of Functions,
Studies in Advanced Math.,
CRC Press, Ann Harbor,
1992. |
MR 1158660 |
Zbl 0804.28001[8]
L. R. JR. FORD-
D. R. FULKERSON,
Flows in networks,
Princeton University Press, Princeton, N.J.,
1962. |
MR 159700 |
Zbl 0106.34802[9]
M. GIAQUINTA-
G. MODICA-
J. SOUČEK,
Cartesian currents in the calculus of variations. I, cartesian currents. II, variational integrals,
Springer-Verlag, Berlin,
1998. |
MR 1645086 |
Zbl 0914.49001[10]
E. GIUSTI,
Minimal surfaces and functions of bounded variation,
Birkhauser, Boston-Basel-Stuttgart,
1984. |
MR 775682 |
Zbl 0545.49018[11]
M. E. GURTIN-
D. POLIGNONE-
J. VIÑALS,
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[12]
F. MORGAN,
Geometric measure theory . A beginner's guide,
Academic Press Inc., San Diego, CA, second ed.,
1995. |
MR 1326605 |
Zbl 0819.49024[13]
S. J. N. MOSCONI-
P. TILLI,
Variational problems with several volume constraints on the level sets, preprint Scuola Norm. Sup. Pisa (
2000). |
Zbl 0995.49003[14]
E. STEPANOV-
P. TILLI,
On the dirichlet problem with several volume constraints on the level sets, preprint Scuola Norm. Sup. Pisa, (
2000). |
MR 1899831 |
Zbl 1022.35010[15]
P. TILLI,
On a constrained variational problem with an arbitrary number of free boundaries,
Interfaces Free Bound.,
2 (
2000), 201-212. |
MR 1760412 |
Zbl 0995.49002