A notion of total variation depending on a metric with discontinuous coefficients

M. Amar; G. Bellettini

Annales de l'I.H.P. Analyse non linéaire (1994)

  • Volume: 11, Issue: 1, page 91-133
  • ISSN: 0294-1449

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Amar, M., and Bellettini, G.. "A notion of total variation depending on a metric with discontinuous coefficients." Annales de l'I.H.P. Analyse non linéaire 11.1 (1994): 91-133. <http://eudml.org/doc/78325>.

@article{Amar1994,
author = {Amar, M., Bellettini, G.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {generalized perimeter; functions of bounded variation; integral functionals; relaxed functional},
language = {eng},
number = {1},
pages = {91-133},
publisher = {Gauthier-Villars},
title = {A notion of total variation depending on a metric with discontinuous coefficients},
url = {http://eudml.org/doc/78325},
volume = {11},
year = {1994},
}

TY - JOUR
AU - Amar, M.
AU - Bellettini, G.
TI - A notion of total variation depending on a metric with discontinuous coefficients
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1994
PB - Gauthier-Villars
VL - 11
IS - 1
SP - 91
EP - 133
LA - eng
KW - generalized perimeter; functions of bounded variation; integral functionals; relaxed functional
UR - http://eudml.org/doc/78325
ER -

References

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  1. [1] G. Alberti, A lusin Type Theorem for Gradients, J. Funct. Anal., Vol. 100 (1), 1991, pp. 110-118. Zbl0752.46025MR1124295
  2. [2] G. Anzellotti, Traces of Bounded Vectorfields and the Divergence Theorm, preprint Univ. Trento, 1983. MR750538
  3. [3] G. Anzellotti, Pairings Between Measures and Bounded Functions and Compensated Compactness, Ann. Mat. Pura e Appl., (4), Vol. 135, 1983, pp. 293-318. Zbl0572.46023MR750538
  4. [4] A.C. Barroso and I. Fonseca, Anisotropic Singular Perturbations. The Vectorial Case, Research Report n. 92-NA-015, Carnegie Mellon University, 1992. Zbl0804.49013
  5. [5] G. Bouchitté and M. Valadier, Integral Representation of Convex Functionals on a Space of Measures, J. Funct. Anal., Vol. 80, 1988, pp. 398-420. Zbl0662.46009MR961907
  6. [6] G. Bouchitté, Singular Perturbations of Variational Problems Arising from a Two-Phase Transition Model, Appl. Math. and Opt., Vol. 21, 1990, pp. 289-315. Zbl0695.49003MR1036589
  7. [7] G. Bouchitté and G. DalMASO, Integral Representation and Relaxation of Convex Local Functionals on BV(Ω), preprint SISSA, April 1991, to appear on Ann. Scuola Norm. Sup., Pisa. MR1267597
  8. [8] H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983. Zbl0511.46001MR697382
  9. [9] H. Busemann and W. Mayer, On the Foundations of Calculus of Variations, Trans. Amer. Math., Soc., Vol. 49, 1941, pp. 173-198. Zbl0024.41703MR3475
  10. [10] H. Busemann, Metric Methods in Finsler Spaces and in the Foundations of Geometry, Ann. of Math. Studies, Vol. 8, Princeton Univ. Press, Princeton, 1942. Zbl0063.00672MR7251
  11. [11] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Longman, Harlow, 1989. Zbl0669.49005MR1020296
  12. [12] G. Buttazzo and G. Dal Maso, Integral Representation and Relaxation of Local Functionals, Nonlinear Analysis, Theory, Methods & Applications, Vol. 9, 1985, pp. 515-532. Zbl0527.49008
  13. [13] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lect. Not. in Math., Vol. 580, Springer-Verlag, Berlin, 1977. Zbl0346.46038MR467310
  14. [14] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, 1989. Zbl0703.49001MR990890
  15. [15] G. Dal Maso, Integral Representation on BV(Ω) of Γ-Limits of Variational Integrals, Manuscripta Math., Vol. 30, 1980, pp. 387-416. Zbl0435.49016MR567216
  16. [16] G. Dal Maso, An Introduction to Γ-Convergence, preprint SISSA, 1992, Trieste, to appear on Birkhäuser. Zbl0816.49001MR1201152
  17. [17] G. Dal Maso and L. Modica, A General Theory of Variational Functionals, in: Topics in Functional Analysis (1980–1981), Quaderni Scuola Norm. Sup. Pisa, Pisa, 1980, pp. 149-221. Zbl0493.49005MR671757
  18. [18] G. De Cecco, Geometria sulle Varietá di Lipschitz, preprint Univ. di Roma "La Sapienza", 1992, pp. 1-38. 
  19. [19] G. De Cecco and G. Palmieri, Distanza Intriseca su una Varietà Riemanniana di Lipschitz, Rend. Sem. Mat. Univ. Pol. Torino, Vol. 46, 1988, pp. 157-170. Zbl0719.53047MR1084564
  20. [20] G. De Cecco and G. Palmieri, Lenght of Curves on Lip manifolds, Rend. Mat. Acc. Lincei, Vol. 1, (9), 1990, pp. 215-221. Zbl0719.53046MR1083250
  21. [21] G. De Cecco and G. Palmieri, Integral Distance on a Lipschitz Riemannian Manifold, Math. Zeitschrift, Vol. 207, 1991, pp. 223-243. Zbl0722.58006MR1109664
  22. [22] E. De Giorgi, Nuovi Teoremi Relativi alle Misure (r-1)-dimensionali in uno Spazio a r Dimensioni, Ricerche Mat. IV, 1955, pp. 95-113. Zbl0066.29903MR74499
  23. [23] E. De Giorgi, Conversazioni di Matematica, Quaderni Dip. Mat. Univ. Lecce, 1988- 1990, Vol. 2, 1990. 
  24. [24] E. De Giorgi, Su Alcuni Problemi Comuni all'Analisi e alla Geometria, Note di Matematica, IX-Suppl., 1989, pp. 59-71. 
  25. [25] E. De Giorgi, Alcuni Problemi Variazionali della Geometria, Conference in onore di G. Aquaro, Bari, Italy, 9 Nov. 1990, 1991, pp. 112-125, Conferenze del Sem. Mat. Bari. 
  26. [26] E. De Giorgi, F. Colombini and L.C. Piccinini, Frontiere Orientate di Misura Minima e Questioni Collegate, Quaderni della Classe di Scienze della Scuola Normale Superiore di Pisa, Pisa, 1972. Zbl0296.49031MR493669
  27. [27] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, New York, 1976. Zbl0322.90046MR463994
  28. [28] H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1968. Zbl0176.00801MR257325
  29. [29] N. Fusco and G. Moscariello, L2-Lower Semicontinuity of Functionals of Quadratic Type, Ann. di Mat. Pura e Appl., Vol. 129, 1981, pp. 305-326. Zbl0483.49008MR648337
  30. [30] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. Zbl0545.49018MR775682
  31. [31] C. Goffman and J. Serrin, Sublinear Functions of Measures and Variational Integrals, Duke Math. J., Vol. 31, 1964, pp. 159-178. Zbl0123.09804MR162902
  32. [32] S. Luckhaus and L. Modica, The Gibbs-Thomson Relation within the Gradient Theory of Phase Transitions, Arch. Rat. Mech. An, Vol. 1, 1989, pp. 71-83. Zbl0681.49012MR1000224
  33. [33] U. Massari and M. Miranda, Minimal Surfaces of Codimension One, North-Holland, New York, 1984. Zbl0565.49030MR795963
  34. [34] V.G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin, 1985. Zbl0692.46023MR817985
  35. [35] M. Miranda, Superfici Cartesiane Generalizzate ed Insiemi di Perimetro Localmente Finito sui Prodotti Cartesiani, Ann. Scuola Norm. Sup., Pisa, Vol. 3, 1964, pp. 515- 542. Zbl0152.24402MR174706
  36. [36] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966. Zbl0142.38701MR202511
  37. [37] J. Neveu, Bases Mathématiques du Calcul des Probabilités, Masson, Paris, 1980. Zbl0203.49901
  38. [38] N. Owen, Non Convex Variational Problems with General Singular Perturbations, Trans. Am. Math. Soc. Vol. 310, 1988, pp. 393-404. Zbl0718.34075
  39. [39] N. Owen and P. Sternberg, Non Convex Variational Problems with Anisotropic Perturbations, Nonlin. Anal., Vol. 16, 1991, pp. 705-719. Zbl0748.49034
  40. [40] C. Pauc, La Méthode Métrique en Calcul des Variations, Hermann, Paris, 1941. Zbl0027.10502MR12736
  41. [41] Yu G. Reshetnyak, Weak Convergence of Completely Additive Vector Functions on a Set , Siberian Math. J., Vol. 9, 1968, pp. 1039-1045. Zbl0176.44402
  42. [42] W. Rinow, Die Innere Geometrie der Metrischen Räume, Springer-Verlag, Berlin, 1961. Zbl0096.16302MR123969
  43. [43] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972. Zbl0193.18401MR1451876
  44. [44] N. Teleman, The Index of Signature Operators on Lipschitz Manifolds, Inst. Hautes Études Sci. Publ. Math., Vol. 58, 1983, pp. 39-78. Zbl0531.58044MR720931
  45. [45] S. Venturini, Derivations of Distance Functions on Rn, preprint, 1993. 
  46. [46] A.I. Vol'pert, The Space BV and Quasilinear Equation, Math. USSR Sbornik, Vol. 2, 1967, pp. 225-267. Zbl0168.07402

Citations in EuDML Documents

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  1. L. Ambrosio, M. Novaga, E. Paolini, Some regularity results for minimal crystals
  2. L. Ambrosio, M. Novaga, E. Paolini, Some regularity results for minimal crystals
  3. Luca Esposito, Nicola Fusco, Cristina Trombetti, A quantitative version of the isoperimetric inequality : the anisotropic case
  4. Nicola Fusco, Virginia De Cicco, Micol Amar, Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands
  5. Angelo Alvino, Vincenzo Ferone, Guido Trombetti, Pierre-Louis Lions, Convex symmetrization and applications
  6. V. Caselles, A. Chambolle, S. Moll, M. Novaga, A characterization of convex calibrable sets in R N with respect to anisotropic norms
  7. Micol Amar, Virginia De Cicco, Nicola Fusco, Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands

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