Geodesic
A Variational Approach to Gradient Flows in Metric Spaces
Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 3, pp. 765-780.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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. 
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Segatti, Antonio. A Variational Approach to Gradient Flows in Metric Spaces. Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 3, pp. 765-780. http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_3_a18/

[1] L. Ambrosio - N. Gigli - G. Savaré, Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. Second edition. | MR

[2] L. Ambrosio - N. Gigli - G. Savaré, Calculus and heat flow in Metric-Measure spaces and applications to spaces with Ricci bounds from below, preprint (2011). | DOI | MR

[3] M. Bardi - I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems & Control: Foundations & Applications. Birkhäuser, 1997. | DOI | MR | Zbl

[4] H. Brézis, 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. | Zbl

[5] H. Brézis - I. Ekeland, 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. | MR | Zbl

[6] S. Conti - M. Ortiz, Minimum principles for the trajectories of systems governed by rate problems, J. Mech. Phys. Solids, 56 (2008), 1885-1904. | DOI | MR | Zbl

[7] M. Crandall - A. Pazy, Semi-groups of nonlinear contractions and dissipative sets, J. Functional Analysis, 3 (1969), 376-418. | DOI | MR | Zbl

[8] E. De Giorgi - A. Marino - M. Tosques, 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. | fulltext EuDML | MR | Zbl

[9] E. De Giorgi, Conjectures concerning some evolution problems, A celebration of John F. Nash, Jr., Duke Math. J., 81 (1996), 255-268. | DOI | MR

[10] Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc., 108, (1994), x+90.

[11] N. Ghoussoub, A theory of anti-selfdual Lagrangians: dynamical case, C. R. Math. Acad. Sci. Paris, 340 (2005), 325-330. | DOI | MR | Zbl

[12] R. Jordan - D. Kinderlehrer - F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. | DOI | MR | Zbl

[13] Y. Kömura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507. | DOI | MR

[14] J. L. Lions, Sur certaines équations paraboliques non linéaires, Bull. Soc. Math. France, 93 (1965), 155-175. | fulltext EuDML | MR | Zbl

[15] J. L. Lions - E. Magenes, 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. | MR | Zbl

[16] A. Mielke - M. Ortiz, A class of minimum principles for characterizing the trajectories of dissipative systems, ESAIM Control Optim. Calc. Var., 14 (2008), 494-516. | fulltext EuDML | DOI | MR | Zbl

[17] A. Mielke - U. Stefanelli, Weighted energy-dissipation functionals for gradient flows, ESAIM Control Optim. Calc. Var., 17 (2011), 52-85. | fulltext EuDML | DOI | MR | Zbl

[18] B. Nayroles, Deux théorèmes de minimum pour certains systèmes dissipatifs, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), Aiv, A1035-A1038. | MR | Zbl

[19] R. H. Nochetto - G. Savaré - C. Verdi, A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math., 53 (2000), 525-589. | DOI | MR | Zbl

[20] F. Otto, The geometry of dissipative evolution: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. | DOI | MR | Zbl

[21] R. Rossi - G. Savaré - A. Segatti - U. Stefanelli, A variational principle for gradient flows in metric spaces, C. R. Acad. Sci. Paris Sér. I Math., 349 (2011), 1225-1228. | DOI | MR | Zbl

[22] R. Rossi - G. Savaré - A. Segatti - U. Stefanelli, Weighted-energy-dissipation functionals for gradient flows in metric spaces, in preparation (2011). | DOI | MR

[23] E. Serra - P. Tilli, 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. | DOI | MR | Zbl

[24] E. N. Spadaro - U. Stefanelli, A variational view at the time-dependent minimal surface equation J. Evol. Equ., 11 (2011), 793-809. | DOI | MR | Zbl

[25] U. Stefanelli, The Brezis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47 (2008), 1615-1642. | DOI | MR | Zbl

[26] U. Stefanelli, The De Giorgi conjecture on elliptic regularization, Math. Models Methods Appl. Sci., 21 (2011), 1377-1394. | DOI | MR | Zbl

[27] A. Visintin, A new approach to evolution, C. R. Acad. Sci. Paris Sér. I Math., 332 (2001), 233-238. | DOI | MR | Zbl