A Variational Approach to Gradient Flows in Metric Spaces

Antonio Segatti

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 3, page 765-780
  • ISSN: 0392-4041

Abstract

top
In this note we report on a new variational principle for Gradient Flows in metric spaces. This new variational formulation consists in a functional defined on entire trajectories whose minimizers converge, in the case in which the energy is geodesically convex, to curves of maximal slope. The key point in the proof is a reformulation of the problem in terms of a dynamic programming principle combined with suitable a priori estimates on the minimizers. The abstract result is applicable to a large class of evolution PDEs, including Fokker Plack equation, drift diffusion and Heat flows in metric-measure spaces.

How to cite

top

Segatti, Antonio. "A Variational Approach to Gradient Flows in Metric Spaces." Bollettino dell'Unione Matematica Italiana 6.3 (2013): 765-780. <http://eudml.org/doc/294052>.

@article{Segatti2013,
abstract = {In this note we report on a new variational principle for Gradient Flows in metric spaces. This new variational formulation consists in a functional defined on entire trajectories whose minimizers converge, in the case in which the energy is geodesically convex, to curves of maximal slope. The key point in the proof is a reformulation of the problem in terms of a dynamic programming principle combined with suitable a priori estimates on the minimizers. The abstract result is applicable to a large class of evolution PDEs, including Fokker Plack equation, drift diffusion and Heat flows in metric-measure spaces.},
author = {Segatti, Antonio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {765-780},
publisher = {Unione Matematica Italiana},
title = {A Variational Approach to Gradient Flows in Metric Spaces},
url = {http://eudml.org/doc/294052},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Segatti, Antonio
TI - A Variational Approach to Gradient Flows in Metric Spaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/10//
PB - Unione Matematica Italiana
VL - 6
IS - 3
SP - 765
EP - 780
AB - In this note we report on a new variational principle for Gradient Flows in metric spaces. This new variational formulation consists in a functional defined on entire trajectories whose minimizers converge, in the case in which the energy is geodesically convex, to curves of maximal slope. The key point in the proof is a reformulation of the problem in terms of a dynamic programming principle combined with suitable a priori estimates on the minimizers. The abstract result is applicable to a large class of evolution PDEs, including Fokker Plack equation, drift diffusion and Heat flows in metric-measure spaces.
LA - eng
UR - http://eudml.org/doc/294052
ER -

References

top
  1. AMBROSIO, L. - GIGLI, N. - SAVARÉ, G., Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. Second edition. MR2401600
  2. AMBROSIO, L. - GIGLI, N. - SAVARÉ, G., Calculus and heat flow in Metric-Measure spaces and applications to spaces with Ricci bounds from below, preprint (2011). MR3152751DOI10.1007/s00222-013-0456-1
  3. BARDI, M. - CAPUZZO-DOLCETTA, I., Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems & Control: Foundations & Applications. Birkhäuser, 1997. Zbl0890.49011MR1484411DOI10.1007/978-0-8176-4755-1
  4. BRÉZIS, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50), North-Holland Publishing Co., Amsterdam, 1973. Zbl0252.47055
  5. BRÉZIS, H. - EKELAND, I., Un principe variationnel associé à certaines équations paraboliques. Le cas indépendant du temps, C. R. Acad. Sci. Paris Sér. A-B, 282, (1976), Aii, A971-A974. Zbl0332.49032MR637214
  6. CONTI, S. - ORTIZ, M., Minimum principles for the trajectories of systems governed by rate problems, J. Mech. Phys. Solids, 56 (2008), 1885-1904. Zbl1162.74369MR2410326DOI10.1016/j.jmps.2007.11.006
  7. CRANDALL, M. - PAZY, A., Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis, 3 (1969), 376-418. Zbl0182.18903MR243383DOI10.1016/0022-1236(69)90032-9
  8. DE GIORGI, E. - MARINO, A. - TOSQUES, M., Problems of evolution in metric spaces and maximal decreasing curve. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68 (1980), 180-187. Zbl0465.47041MR636814
  9. DE GIORGI, E., Conjectures concerning some evolution problems, A celebration of John F. Nash, Jr., Duke Math. J., 81 (1996), 255-268. MR1395405DOI10.1215/S0012-7094-96-08114-4
  10. Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc., 108, (1994), x+90. 
  11. GHOUSSOUB, N., A theory of anti-selfdual Lagrangians: dynamical case, C. R. Math. Acad. Sci. Paris, 340 (2005), 325-330. Zbl1070.49029MR2121900DOI10.1016/j.crma.2004.12.008
  12. JORDAN, R. - KINDERLEHRER, D. - OTTO, F., The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. Zbl0915.35120MR1617171DOI10.1137/S0036141096303359
  13. KÖMURA, Y., Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507. MR216342DOI10.2969/jmsj/01940493
  14. LIONS, J. L., Sur certaines équations paraboliques non linéaires, Bull. Soc. Math. France, 93 (1965), 155-175. Zbl0132.10601MR194760
  15. LIONS, J. L. - MAGENES, E., Non-homogeneous boundary value problems and applications. Vol. I, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York, 1972. Zbl0223.35039MR350177
  16. MIELKE, A. - ORTIZ, M., A class of minimum principles for characterizing the trajectories of dissipative systems, ESAIM Control Optim. Calc. Var., 14 (2008), 494-516. Zbl1357.49043MR2434063DOI10.1051/cocv:2007064
  17. MIELKE, A. - STEFANELLI, U., Weighted energy-dissipation functionals for gradient flows, ESAIM Control Optim. Calc. Var., 17 (2011), 52-85. Zbl1218.35007MR2775186DOI10.1051/cocv/2009043
  18. NAYROLES, B., Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), Aiv, A1035-A1038. Zbl0345.73037MR418609
  19. NOCHETTO, R. H. - SAVARÉ, G. - VERDI, C., A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math., 53 (2000), 525-589. Zbl1021.65047MR1737503DOI10.1002/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M
  20. OTTO, F., The geometry of dissipative evolution: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. Zbl0984.35089MR1842429DOI10.1081/PDE-100002243
  21. ROSSI, R. - SAVARÉ, G. - SEGATTI, A. - STEFANELLI, U., A variational principle for gradient flows in metric spaces, C. R. Acad. Sci. Paris Sér. I Math., 349 (2011), 1225-1228. Zbl1236.49065MR2861989DOI10.1016/j.crma.2011.11.002
  22. ROSSI, R. - SAVARÉ, G. - SEGATTI, A. - STEFANELLI, U., Weighted-energy-dissipation functionals for gradient flows in metric spaces, in preparation (2011). MR2763076DOI10.1016/j.matpur.2010.10.011
  23. SERRA, E. - TILLI, P., Nonlinear wave equations as limits of convex minimization problems: proof of a conjecture by De Giorgi, Ann. of Math. (2), 175 (2012), 3, 1551-1574. Zbl1251.49019MR2912711DOI10.4007/annals.2012.175.3.11
  24. SPADARO, E. N. - STEFANELLI, U., A variational view at the time-dependent minimal surface equationJ. Evol. Equ., 11 (2011), 793-809. Zbl1246.35117MR2861306DOI10.1007/s00028-011-0111-5
  25. STEFANELLI, U., The Brezis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47 (2008), 1615-1642. Zbl1194.35214MR2425653DOI10.1137/070684574
  26. STEFANELLI, U., The De Giorgi conjecture on elliptic regularization, Math. Models Methods Appl. Sci., 21 (2011), 1377-1394. Zbl1228.35023MR2819200DOI10.1142/S0218202511005350
  27. VISINTIN, A., A new approach to evolution, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), 233-238. Zbl0977.35142MR1817368DOI10.1016/S0764-4442(00)01825-5

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.