Dimension reduction for −Δ1
Maria Emilia Amendola; Giuliano Gargiulo; Elvira Zappale
ESAIM: Control, Optimisation and Calculus of Variations (2014)
- Volume: 20, Issue: 1, page 42-77
- ISSN: 1292-8119
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topAmendola, Maria Emilia, Gargiulo, Giuliano, and Zappale, Elvira. "Dimension reduction for −Δ1." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 42-77. <http://eudml.org/doc/272817>.
@article{Amendola2014,
abstract = {A 3D-2D dimension reduction for −Δ1 is obtained. A power law approximation from −Δp as p → 1 in terms of Γ-convergence, duality and asymptotics for least gradient functions has also been provided.},
author = {Amendola, Maria Emilia, Gargiulo, Giuliano, Zappale, Elvira},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {1–Laplacian; Γ–convergence; least gradient functions; dimension reduction; duality; 1-Laplacian; -convergence},
language = {eng},
number = {1},
pages = {42-77},
publisher = {EDP-Sciences},
title = {Dimension reduction for −Δ1},
url = {http://eudml.org/doc/272817},
volume = {20},
year = {2014},
}
TY - JOUR
AU - Amendola, Maria Emilia
AU - Gargiulo, Giuliano
AU - Zappale, Elvira
TI - Dimension reduction for −Δ1
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 1
SP - 42
EP - 77
AB - A 3D-2D dimension reduction for −Δ1 is obtained. A power law approximation from −Δp as p → 1 in terms of Γ-convergence, duality and asymptotics for least gradient functions has also been provided.
LA - eng
KW - 1–Laplacian; Γ–convergence; least gradient functions; dimension reduction; duality; 1-Laplacian; -convergence
UR - http://eudml.org/doc/272817
ER -
References
top- [1] E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string. J. Elast.25 (1991) 137–148. Zbl0734.73094MR1111364
- [2] L. Ambrosio and G. Dal Maso, On the relaxation in BV(Ω;Rm) of quasi–convex integrals. J. Funct. Anal.109 (1992) 76–97. Zbl0769.49009MR1183605
- [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Oxford: Clarendon Press (2000). Zbl0957.49001MR1857292
- [4] R.A. Adams, Sobolev Spaces. Academic Press, New York (1975). Zbl1098.46001MR450957
- [5] G. Anzellotti, Pairing between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl.135 (1983) 293–318. Zbl0572.46023MR750538
- [6] A. Braides and I. Fonseca, Brittle thin films. Appl. Math. Optim.44 (2001) 299–323. Zbl0999.49012MR1851742
- [7] J.F. Babadjian, E. Zappale and H. Zorgati, Dimensional reduction for energies with linear growth involving the bending moment. J. Math. Pures Appl.90 (2008) 520–549. Zbl1153.49017MR2472892
- [8] M. Bocea and V. Nesi, Γ–convergence of power-law functionals, variational principles in L∞, and applications. SIAM J. Math. Anal.39 (2008) 1550–1576. Zbl1166.35300MR2377289
- [9] H. Brezis, Analisi Funzionale. Liguori, Napoli (1986).
- [10] G. Dal Maso, An introduction to Γ–convergence. Progress Nonlinear Differ. Equ. Appl. Birkhäuser Boston, Inc., Boston, MA (1983). Zbl0816.49001
- [11] R. De Arcangelis and C. Trombetti, On the relaxation of some classes of Dirichlet minimum problems. Commun. Partial Differ. Eqs.24 (1999) 975–1006. Zbl0928.49014MR1680889
- [12] E. De Giorgi, G. Letta, Une notion generale de convergence faible pour des fonctions croissantes d’ensemble. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 4 (1977) 61–99. Zbl0405.28008MR466479
- [13] F. Demengel, On Some Nonlinear Partial Differential Equations involving the 1–Laplacian and Critical Sobolev Exponent. ESAIM: COCV 4 (1999) 667–686. Zbl0939.35070MR1746172
- [14] F. Demengel, Théorèmes d’existence pour des equations avec l’opérateur 1-Laplacien, première valeur propre pour − Δ1. (French) [Some existence results for partial differential equations involving the 1-Laplacian: first eigenvalue for − Δ1]. C. R. Math. Acad. Sci. Paris 334 (2002) 1071–1076. Zbl1142.35408MR1911649
- [15] F. Demengel, Functions locally almost 1–harmonic. Appl. Anal.83 (2004) 865–896. Zbl1135.35333MR2083734
- [16] F. Demengel, On some nonlinear equation involving the 1–Laplacian and trace map inequalities. Nonlinear Anal.47 (2002) 1151–1163. Zbl1016.49001MR1880578
- [17] I. Ekeland and R. Temam, Convex analysis and variational problems. North-Holland, Amsterdam (1976). Zbl0322.90046MR463994
- [18] I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in BV(Ω,RN) for integrands f(x,u,∇u). Arch. Ration. Mech. Anal.123 (1993) 1–49. Zbl0788.49039MR1218685
- [19] M. Giaquinta, G. Modica and J. Soucek, Functionals with linear growth in the Calculus of Variations. Comment. Math. Univ. Carolin.20 (1979) 143–156. Zbl0409.49006MR526154
- [20] C. Goffman and J. Serrin, Sublinear functions of Measures and Variational Integrals. Duke Math. J.31 (1964) 159–178. Zbl0123.09804MR162902
- [21] E. Giusti, Minimal surfaces and functions of bounded variation. Birkhauser (1977). Zbl0402.49033MR638362
- [22] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford, New York, Tokyo, Clarendon Press (1993). Zbl0780.31001MR1207810
- [23] P. Juutinen, p–harmonic approximation of functions of least gradient. Indiana Univ. Math. J. 54 (2005) 1015–1029. Zbl1100.49025MR2164415
- [24] B. Kawhol, Variations on the p–Laplacian, in edited by D. Bonheure, P. Takac. Nonlinear Elliptic Partial Differ. Equ. Contemporary Math. 540 (2011) 35–46. Zbl1241.35109MR2841298
- [25] R. Kohn and R. Temam, Dual spaces of Stresses and Strains, with Applications to Hencky Plasticity. Appl. Math. Optim.10 (1983) 1–35. Zbl0532.73039MR701898
- [26] H. Le Dret, and A. Raoult, The nonlinear membrane model as a variational limit of nonlinear three–dimensional elasticity. J. Math. Pures Appl.74 (1995) 549–578. Zbl0847.73025MR1365259
- [27] P. Lindqvist, On the Equation div( | ∇u | p − 2∇u) + λ | u | p − 2u = 0. Proc. Amer. Math. Soc. 109 (1990) 157–164. Zbl0714.35029MR1007505
- [28] S. Monsurró and E. Zappale, On the relaxation and homogenization of some classes of variational problems with mixed boundary conditions. Rev. Roum. Math. Pures Appl.51 (2006) 345–363. Zbl1120.49012MR2275828
- [29] P. Sternberg, G. Williams and W. P. Ziemer, Existence, uniqueness, and regularity for functions of least gradient. J. Reine Angew. Math. 430 (1992) 3560. Zbl0756.49021MR1172906
- [30] P. Sternberg and W.P. Ziemer, The Dirichlet problem for functions of least gradient. In Degenerate diffusions (Minneapolis, MN, 1991). In vol. 47 of IMA Vol. Math. Appl. Springer, New York (1993) 197–214. Zbl0818.49024MR1246349
- [31] E. Zappale, On the homogenization of Dirichlet Minimum Problems. Ricerche di Matematica LI (2002) 61–92. Zbl1146.35392MR1961839
- [32] W.P. Ziemer, Weakly differentiable functions. In vol. 120 of Graduate Texts in Math. Springer, Berlin (1989). Zbl0692.46022MR1014685
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