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Dimension reduction for −Δ1

Published online by Cambridge University Press:  03 September 2013

Maria Emilia Amendola ,
Giuliano Gargiulo  and
Elvira Zappale
Maria Emilia Amendola
Affiliation:
Dipartimento di Matematica, Universita’ degli Studi di Salerno, via Ponte Don Melillo, 84084 Fisciano (SA), Italy. emamendola@unisa.it
Giuliano Gargiulo
Affiliation:
DSBGA, Universita’ del Sannio, Benevento Italy; ggargiul@unisannio.it
Elvira Zappale
Affiliation:
Dipartimento di Ingegneria Industriale, Universita’ degli Studi di Salerno, via Ponte Don Melillo, 84084 Fisciano (SA), Italy; ezappale@unisa.it
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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.

Keywords

1–LaplacianΓ–convergenceleast gradient functionsdimension reductionduality
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 42 - 77
Copyright
© EDP Sciences, SMAI, 2013

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References

Acerbi, E., Buttazzo, G. and Percivale, D., A variational definition of the strain energy for an elastic string. J. Elast. 25 (1991) 137–148. Google Scholar
Ambrosio, L. and Dal Maso, G., On the relaxation in BV(Ω;Rm) of quasi–convex integrals. J. Funct. Anal. 109 (1992) 76–97. Google Scholar
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Oxford: Clarendon Press (2000).
R.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
Anzellotti, G., Pairing between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 135 (1983) 293–318. Google Scholar
Braides, A. and Fonseca, I., Brittle thin films. Appl. Math. Optim. 44 (2001) 299–323. Google Scholar
Babadjian, J.F., Zappale, E. and Zorgati, H., Dimensional reduction for energies with linear growth involving the bending moment. J. Math. Pures Appl. 90 (2008) 520–549. Google Scholar
Bocea, M. and Nesi, V., Γ–convergence of power-law functionals, variational principles in L ∞, and applications. SIAM J. Math. Anal. 39 (2008) 1550–1576. Google Scholar
H. Brezis, Analisi Funzionale. Liguori, Napoli (1986).
G. Dal Maso, An introduction to Γ–convergence. Progress Nonlinear Differ. Equ. Appl. Birkhäuser Boston, Inc., Boston, MA (1983).
De Arcangelis, R. and Trombetti, C., On the relaxation of some classes of Dirichlet minimum problems. Commun. Partial Differ. Eqs. 24 (1999) 975–1006. Google Scholar
De Giorgi, E., Letta, G., 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. Google Scholar
Demengel, F., On Some Nonlinear Partial Differential Equations involving the 1–Laplacian and Critical Sobolev Exponent. ESAIM: COCV 4 (1999) 667–686. Google Scholar
Demengel, F., 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. Google Scholar
Demengel, F., Functions locally almost 1–harmonic. Appl. Anal. 83 (2004) 865–896. Google Scholar
Demengel, F., On some nonlinear equation involving the 1–Laplacian and trace map inequalities. Nonlinear Anal. 47 (2002) 1151–1163. Google Scholar
I. Ekeland and R. Temam, Convex analysis and variational problems. North-Holland, Amsterdam (1976).
Fonseca, I. and Müller, S., Relaxation of quasiconvex functionals in BV(Ω,RN) for integrands f(x,u,∇u). Arch. Ration. Mech. Anal. 123 (1993) 1–49. Google Scholar
Giaquinta, M., Modica, G. and Soucek, J., Functionals with linear growth in the Calculus of Variations. Comment. Math. Univ. Carolin. 20 (1979) 143–156. Google Scholar
Goffman, C. and Serrin, J., Sublinear functions of Measures and Variational Integrals. Duke Math. J. 31 (1964) 159–178. Google Scholar
E. Giusti, Minimal surfaces and functions of bounded variation. Birkhauser (1977).
J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford, New York, Tokyo, Clarendon Press (1993).
Juutinen, P., p–harmonic approximation of functions of least gradient. Indiana Univ. Math. J. 54 (2005) 1015–1029. Google Scholar
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.
Kohn, R. and Temam, R., Dual spaces of Stresses and Strains, with Applications to Hencky Plasticity. Appl. Math. Optim. 10 (1983) 1–35. Google Scholar
Le Dret, H., and Raoult, A., The nonlinear membrane model as a variational limit of nonlinear three–dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549–578. Google Scholar
Lindqvist, P., On the Equation div( | ∇u | p − 2∇u) + λ | u | p − 2u = 0. Proc. Amer. Math. Soc. 109 (1990) 157–164. Google Scholar
Monsurró, S. and Zappale, E., On the relaxation and homogenization of some classes of variational problems with mixed boundary conditions. Rev. Roum. Math. Pures Appl. 51 (2006) 345–363. Google Scholar
Sternberg, P., Williams, G. and Ziemer, W. P., Existence, uniqueness, and regularity for functions of least gradient. J. Reine Angew. Math. 430 (1992) 3560. Google Scholar
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.
E. Zappale, On the homogenization of Dirichlet Minimum Problems. Ricerche di Matematica LI (2002) 61–92.
W.P. Ziemer, Weakly differentiable functions. In vol. 120 of Graduate Texts in Math. Springer, Berlin (1989).