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.
Mots clés : 1-Laplacian, Γ-convergence, least gradient functions, dimension reduction, duality
@article{COCV_2014__20_1_42_0,
author = {Amendola, Maria Emilia and Gargiulo, Giuliano and Zappale, Elvira},
title = {Dimension reduction for $-\Delta _1$},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {42--77},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
doi = {10.1051/cocv/2013053},
zbl = {1288.35266},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2013053/}
}
TY - JOUR AU - Amendola, Maria Emilia AU - Gargiulo, Giuliano AU - Zappale, Elvira TI - Dimension reduction for $-\Delta _1$ JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 42 EP - 77 VL - 20 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2013053/ DO - 10.1051/cocv/2013053 LA - en ID - COCV_2014__20_1_42_0 ER -
%0 Journal Article %A Amendola, Maria Emilia %A Gargiulo, Giuliano %A Zappale, Elvira %T Dimension reduction for $-\Delta _1$ %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 42-77 %V 20 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2013053/ %R 10.1051/cocv/2013053 %G en %F COCV_2014__20_1_42_0
Amendola, Maria Emilia; Gargiulo, Giuliano; Zappale, Elvira. Dimension reduction for $-\Delta _1$. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 42-77. doi : 10.1051/cocv/2013053. http://www.numdam.org/articles/10.1051/cocv/2013053/
[1] , and , A variational definition of the strain energy for an elastic string. J. Elast. 25 (1991) 137-148. | MR | Zbl
[2] and , On the relaxation in BV(Ω;Rm) of quasi-convex integrals. J. Funct. Anal. 109 (1992) 76-97. | MR | Zbl
[3] , and , Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Oxford: Clarendon Press (2000). | MR | Zbl
[4] , Sobolev Spaces. Academic Press, New York (1975). | MR | Zbl
[5] , Pairing between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 135 (1983) 293-318. | MR | Zbl
[6] and , Brittle thin films. Appl. Math. Optim. 44 (2001) 299-323. | MR | Zbl
[7] , and , Dimensional reduction for energies with linear growth involving the bending moment. J. Math. Pures Appl. 90 (2008) 520-549. | MR | Zbl
[8] and , Γ-convergence of power-law functionals, variational principles in L∞, and applications. SIAM J. Math. Anal. 39 (2008) 1550-1576. | MR | Zbl
[9] , Analisi Funzionale. Liguori, Napoli (1986).
[10] , An introduction to Γ-convergence. Progress Nonlinear Differ. Equ. Appl. Birkhäuser Boston, Inc., Boston, MA (1983). | Zbl
[11] and , On the relaxation of some classes of Dirichlet minimum problems. Commun. Partial Differ. Eqs. 24 (1999) 975-1006. | MR | Zbl
[12] , , 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. | Numdam | MR | Zbl
[13] , On Some Nonlinear Partial Differential Equations involving the 1-Laplacian and Critical Sobolev Exponent. ESAIM: COCV 4 (1999) 667-686. | Numdam | MR | Zbl
[14] , 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. | MR | Zbl
[15] , Functions locally almost 1-harmonic. Appl. Anal. 83 (2004) 865-896. | MR | Zbl
[16] , On some nonlinear equation involving the 1-Laplacian and trace map inequalities. Nonlinear Anal. 47 (2002) 1151-1163. | MR | Zbl
[17] and , Convex analysis and variational problems. North-Holland, Amsterdam (1976). | MR | Zbl
[18] and , Relaxation of quasiconvex functionals in BV(Ω,RN) for integrands f(x,u,∇u). Arch. Ration. Mech. Anal. 123 (1993) 1-49. | MR | Zbl
[19] , and , Functionals with linear growth in the Calculus of Variations. Comment. Math. Univ. Carolin. 20 (1979) 143-156. | MR | Zbl
[20] and , Sublinear functions of Measures and Variational Integrals. Duke Math. J. 31 (1964) 159-178. | MR | Zbl
[21] , Minimal surfaces and functions of bounded variation. Birkhauser (1977). | MR | Zbl
[22] , and , Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford, New York, Tokyo, Clarendon Press (1993). | MR | Zbl
[23] P. Juutinen, p-harmonic approximation of functions of least gradient. Indiana Univ. Math. J. 54 (2005) 1015-1029. | MR | Zbl
[24] , Variations on the p-Laplacian, in edited by D. Bonheure, P. Takac. Nonlinear Elliptic Partial Differ. Equ. Contemporary Math. 540 (2011) 35-46. | MR
[25] and , Dual spaces of Stresses and Strains, with Applications to Hencky Plasticity. Appl. Math. Optim. 10 (1983) 1-35. | MR | Zbl
[26] , and , The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549-578. | MR | Zbl
[27] , On the Equation div( | ∇u | p − 2∇u) + λ | u | p − 2u = 0. Proc. Amer. Math. Soc. 109 (1990) 157-164. | MR | Zbl
[28] and , On the relaxation and homogenization of some classes of variational problems with mixed boundary conditions. Rev. Roum. Math. Pures Appl. 51 (2006) 345-363. | MR | Zbl
[29] , and , Existence, uniqueness, and regularity for functions of least gradient. J. Reine Angew. Math. 430 (1992) 3560. | MR | Zbl
[30] and , 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. | MR | Zbl
[31] , On the homogenization of Dirichlet Minimum Problems. Ricerche di Matematica LI (2002) 61-92. | MR | Zbl
[32] , Weakly differentiable functions. In vol. 120 of Graduate Texts in Math. Springer, Berlin (1989). | MR | Zbl
Cité par Sources :
