Ansini, N. and Iosifescu, O.:
Approximation of Anisotropic Perimeter Functionals by Homogenization
Bollettino dell'Unione Matematica Italiana Serie 9 3 (2010), fasc. n.1, p. 149-168, (English)
pdf (381 Kb), djvu (174 Kb). | MR 2605917 | Zbl 1196.49032
Sunto
We show that all anisotropic perimeter functionals of the form $\int_{\partial^{\star}E \cap \Omega} \varphi(\nu_{E}) \, d\mathcal{H}^{n-1}$ ($\varphi$ convex and positively homogeneous of degree one) can be approximated in the sense of $\Gamma$-convergence by (limits of) isotropic but inhomogeneous perimeter functionals of the form $\int_{\partial^{\star}E \cap \Omega} a(x/\epsilon) \, d\mathcal{H}^{n-1}$ ($a$ periodic).
Referenze Bibliografiche
[2]
F. J. ALMGREN -
J. E. TAYLOR,
Flat flow is motion by crystalline curvature for curves with crystalline energies,
J. Differential Geom.,
42 (
1995), 1-22. |
MR 1350693 |
Zbl 0867.58020[3]
L. AMBROSIO -
A. BRAIDES,
Functionals defined on partitions of sets of finite perimeter I: integral representation and $\Gamma$-convergence,
J. Math. Pures Appl.,
69 (
1990), 285-305. |
MR 1070481 |
Zbl 0676.49028[4]
L. AMBROSIO -
A. BRAIDES,
Functionals defined on partitions of sets of finite perimeter II: semicontinuity, relaxation and homogenization,
J. Math. Pures Appl.,
69 (
1990), 307-333. |
MR 1070482 |
Zbl 0676.49029[5]
L. AMBROSIO -
N. FUSCO -
D. PALLARA,
Functions of bounded variation and free discontinuity problems,
Oxford University Press,
2000. |
MR 1857292 |
Zbl 0957.49001[6]
N. ANSINI -
A. BRAIDES -
V. CHIADÒ PIAT,
Gradient theory of phase transitions in composite media,
Proc. Royal Soc. Edinburgh A,
133 (
2003), 265-296. |
fulltext (doi) |
MR 1969814 |
Zbl 1031.49021[7]
G. BELLETTINI -
R. GOGLIONE -
M. NOVAGA,
Approximation to driven motion by crystalline curvature in two dimensions,
Adv. Math. Sci. and Appl.,
10 (
2000), 467-493. |
MR 1769163 |
Zbl 0979.53075[8]
G. BELLETTINI -
M. NOVAGA,
Approximation and comparison for nonsmooth anisotropic motion by mean curvature in $\mathbb{R}^{n}$,
Math. Mod. Meth. Appl. Sc.,
10 (
2000), 1-10. |
fulltext (doi) |
MR 1749692 |
Zbl 1016.53048[9]
G. BOUCHITTÉ -
I. FONSECA -
G. LEONI -
L. MASCARENHAS,
A global method for relaxation in $W^{1,p}$ and in $SBV_{p}$,
Arch. Rational Mech. Anal.,
165, 3 (
2002), 187-242. |
fulltext (doi) |
MR 1941478[11]
A. BRAIDES -
G. BUTTAZZO -
I. FRAGALÀ,
Riemannian approximation of Finsler metrics,
Asymptot. Anal.,
31 (
2002), 117-187. |
MR 1938603 |
Zbl 1032.49020[12]
A. BRAIDES -
V. CHIADÒ PIAT,
Integral representation results for functionals defined on $SBV(\Omega,\mathbb{R}^{m})$,
J. Math. Pures Appl.,
75 (
1996), 595-626. |
MR 1423049 |
Zbl 0880.49010[13]
A. BRAIDES -
A. DEFRANCESCHI,
Homogenization of Multiple Integrals,
Oxford University Press, Oxford,
1998. |
MR 1684713 |
Zbl 0911.49010[16]
A. DAVINI,
On the relaxation of a class of functionals defined on Riemannian distances,
J. Convex Anal.,
12 (
2005), 113-130. |
MR 2135800 |
Zbl 1078.49012[17]
A. DAVINI,
Smooth approximation of weak Finsler metrics,
Differential Integral Equations,
18 (
2005), 509-530. |
MR 2136977 |
Zbl 1212.41092[20]
L. MODICA -
S. MORTOLA,
Un esempio di $\Gamma$-convergenza,
Boll. Un. Mat. Ital.,
14-B (
1977), 285-299. |
MR 445362[21]
L. MODICA,
The gradient theory of phase transitions and the minimal interface criterion,
Arch. Rational Mech. Anal.,
98 (
1987), 123-142. |
fulltext (doi) |
MR 866718 |
Zbl 0616.76004[23]
J. E. TAYLOR,
Constructions and conjectures in crystalline nondifferential geometry,
Differential Geometry, eds.
B. Lawson and
K. Tanenblat,
Pitman Monographs in Pure and Applied Math.,
52 (
Pitman,
1991), 321-336. |
fulltext (doi) |
MR 1173051 |
Zbl 0725.53011[24] J. E. TAYLOR, Mean curvature and weighted mean curvature II, Acta Metall. Mater., 40 (1992) 1475-1485.
[25]
J. E. TAYLOR,
Motion of curves by crystalline curvature, including triple junctions and boundary points,
Proc. Symp. Pure Math.,
54 (
1993) 417-438. |
fulltext (doi) |
MR 1216599 |
Zbl 0823.49028
Con il contributo del Ministero per i Beni e le Attività Culturali