Evolution Problems and Minimizing Movements

Ugo Gianazza; Massimo Gobbino; Giuseppe Savarè

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1994)

  • Volume: 5, Issue: 4, page 289-296
  • ISSN: 1120-6330

Abstract

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We recall the definition of Minimizing Movements, suggested by E. De Giorgi, and we consider some applications to evolution problems. With regards to ordinary differential equations, we prove in particular a generalization of maximal slope curves theory to arbitrary metric spaces. On the other hand we present a unifying framework in which some recent conjectures about partial differential equations can be treated and solved. At the end we consider some open problems.

How to cite

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Gianazza, Ugo, Gobbino, Massimo, and Savarè, Giuseppe. "Evolution Problems and Minimizing Movements." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 5.4 (1994): 289-296. <http://eudml.org/doc/244234>.

@article{Gianazza1994,
abstract = {We recall the definition of Minimizing Movements, suggested by E. De Giorgi, and we consider some applications to evolution problems. With regards to ordinary differential equations, we prove in particular a generalization of maximal slope curves theory to arbitrary metric spaces. On the other hand we present a unifying framework in which some recent conjectures about partial differential equations can be treated and solved. At the end we consider some open problems.},
author = {Gianazza, Ugo, Gobbino, Massimo, Savarè, Giuseppe},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Minimizing Movements; Variational evolution problems; Gradient flow; Maximal slope; variational evolution problems; Cauchy problem; gradient type differential equations; maximal slope curves; parabolic equations; minimizing movements},
language = {eng},
month = {12},
number = {4},
pages = {289-296},
publisher = {Accademia Nazionale dei Lincei},
title = {Evolution Problems and Minimizing Movements},
url = {http://eudml.org/doc/244234},
volume = {5},
year = {1994},
}

TY - JOUR
AU - Gianazza, Ugo
AU - Gobbino, Massimo
AU - Savarè, Giuseppe
TI - Evolution Problems and Minimizing Movements
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1994/12//
PB - Accademia Nazionale dei Lincei
VL - 5
IS - 4
SP - 289
EP - 296
AB - We recall the definition of Minimizing Movements, suggested by E. De Giorgi, and we consider some applications to evolution problems. With regards to ordinary differential equations, we prove in particular a generalization of maximal slope curves theory to arbitrary metric spaces. On the other hand we present a unifying framework in which some recent conjectures about partial differential equations can be treated and solved. At the end we consider some open problems.
LA - eng
KW - Minimizing Movements; Variational evolution problems; Gradient flow; Maximal slope; variational evolution problems; Cauchy problem; gradient type differential equations; maximal slope curves; parabolic equations; minimizing movements
UR - http://eudml.org/doc/244234
ER -

References

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  1. BREZIS, H., Opérateurs maximaux monotones et semigroups de contraction dans les espaces de Hilbert. North-Holland Mathematics Studies, n. 5, Notas de Matematica, (50), Amsterdam-London1973. Zbl0252.47055MR348562
  2. CRANDALL, M. G., An introduction to evolution governed by accretive operators. Dynamical Systems - An International Symposium, Academic Press, 1976, 131-165. Zbl0339.35049MR636953
  3. CRANDALL, M. G. - LIGGET, J. M., Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math., 93, 1971, 265-298. Zbl0226.47038MR287357
  4. DE GIORGI, E., Some conjectures on flow by mean curvature. In: M. L. BENEVENTO - T. BRUNO - C. SBORDONE (eds.), Methods of Real Analysis and PDEs. Liguori, Napoli1990. Zbl0840.35042
  5. DE GIORGI, E., Movimenti Minimizzanti. Proceedings of the Conference on Aspetti e problemi della Matematica oggi. Lecce1992. 
  6. DE GIORGI, E., New Problems on Minimizing Movements. In: Boundary Value Problems for PDEs and Applications. Masson, 1993, 81-98. Zbl0851.35052MR1260440
  7. DE GIORGI, E. - MARINO, A. - TOSQUES, M., Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti Acc. Lincei Rend. fis., s. 8, v. 68, 1980, 180-187. Zbl0465.47041MR636814
  8. DE GIORGI, E. - MARINO, A. - TOSQUES, M., Funzioni p , q -convesse. Atti Acc. Lincei Rend. fis., s. 8, v. 73, 1982, 6-14. Zbl0521.49011MR726279
  9. DEGIOVANNI, M. - MARINO, A. - TOSQUES, M., Evolution equations associated with p , q -convex functions and p , q -monotone operators. Ric. Mat., XXXIII, 1, 1984, 81-112. Zbl0582.49005MR795157
  10. DENKOWSKI, Z. - MIGÓRSKI, S. - MORTOLA, S., Differential Inclusions and Minimizing Movements. SNS, Pisa1993, preprints di Matematica, 36. Zbl0954.49002
  11. GIANAZZA, U. - SAVARÈ, G., Some results on Minimizing Movements. Istituto di Analisi Numerica del CNR, Pavia1994, preprint n. 914. Zbl0862.49016MR1327457
  12. MARINO, A. - SACCON, C. - TOSQUES, M., Curves of maximal slope and parabolic variational inequalities on non-convex constraints. Ann. Sc. Normale Sup., 4, XVI, 1989, 281-330. Zbl0699.49015MR1041899

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