Geodesic
Approximation of Anisotropic Perimeter Functionals by Homogenization
Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 1, pp. 149-168.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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). 
@article{BUMI_2010_9_3_1_a6,
     author = {Ansini, N. and Iosifescu, O.},
     title = {Approximation of {Anisotropic} {Perimeter} {Functionals} by {Homogenization}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {149--168},
     publisher = {mathdoc},
     volume = {Ser. 9, 3},
     number = {1},
     year = {2010},
     zbl = {1196.49032},
     mrnumber = {2605917},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_1_a6/}
}
TY  - JOUR
AU  - Ansini, N.
AU  - Iosifescu, O.
TI  - Approximation of Anisotropic Perimeter Functionals by Homogenization
JO  - Bollettino della Unione matematica italiana
PY  - 2010
SP  - 149
EP  - 168
VL  - 3
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_1_a6/
LA  - en
ID  - BUMI_2010_9_3_1_a6
ER  - 
%0 Journal Article
%A Ansini, N.
%A Iosifescu, O.
%T Approximation of Anisotropic Perimeter Functionals by Homogenization
%J Bollettino della Unione matematica italiana
%D 2010
%P 149-168
%V 3
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_1_a6/
%G en
%F BUMI_2010_9_3_1_a6
Ansini, N.; Iosifescu, O. Approximation of Anisotropic Perimeter Functionals by Homogenization. Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 1, pp. 149-168. http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_1_a6/

[1] E. Acerbi - G. Buttazzo, On the limits of periodic Riemannian metrics, J. Anal. Math., 43 (1984), 183-201. | DOI | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[5] L. Ambrosio - N. Fusco - D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, 2000. | MR | Zbl

[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. | DOI | MR | Zbl

[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 | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR

[10] A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics (Springer Verlag, Berlin, 1998). | DOI | MR | Zbl

[11] A. Braides - G. Buttazzo - I. Fragalà, Riemannian approximation of Finsler metrics, Asymptot. Anal., 31 (2002), 117-187. | MR | Zbl

[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 | Zbl

[13] A. Braides - A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, Oxford, 1998. | MR | Zbl

[14] A. Braides - M. Maslennikov - L. Sigalotti, Homogenization by blow-up. Applicable Anal., 87 (2008), 1341-1356. | DOI | MR | Zbl

[15] G. Dal Maso, An Introduction to $\Gamma$-convergence, Birkhäuser, Boston, 1993. | DOI | MR | Zbl

[16] A. Davini, On the relaxation of a class of functionals defined on Riemannian distances, J. Convex Anal., 12 (2005), 113-130. | MR | Zbl

[17] A. Davini, Smooth approximation of weak Finsler metrics, Differential Integral Equations, 18 (2005), 509-530. | MR | Zbl

[18] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. | DOI | MR | Zbl

[19] H. Federer, Geometric Measure Theory, Springer Verlag, Berlin, 1968. | MR | Zbl

[20] L. Modica - S. Mortola, Un esempio di $\Gamma$-convergenza, Boll. Un. Mat. Ital., 14-B (1977), 285-299. | MR

[21] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142. | DOI | MR | Zbl

[22] J. E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc., 84 (1978), 568-588. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl