Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T16:49:53.529Z Has data issue: false hasContentIssue false

Gradient flows of non convex functionals in Hilbert spaces and applications

Published online by Cambridge University Press:  20 June 2006

Riccarda Rossi
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia. Via Ferrata, 1 – 27100 Pavia, Italy; [email protected]; [email protected]
Giuseppe Savaré
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia. Via Ferrata, 1 – 27100 Pavia, Italy; [email protected]; [email protected]
Get access

Abstract

This paper addresses the Cauchy problem for thegradient flow equation in a Hilbert space  $\mathcal{H}$ \[\begin{cases}u'(t)+ \partial_{\ell}\phi(u(t))\ni f(t)&\text{{\it a.e.}\ in }(0,T),u(0)=u_0, \end{cases}\] where $\phi: \mathcal{H} \to (-\infty,+\infty]$ is a proper,lower semicontinuous functional which is not supposed to be a(smooth perturbation of a) convex functional and $\partial_{\ell}\phi$ is(a suitable limiting version of) its subdifferential.We will present some new existence results for the solutions of theequation by exploiting a variational approximationtechnique, featuring some ideas from the theory of Minimizing Movementsand of Young measures.
Our analysisis also motivated by some models describing phase transitionsphenomena, leading tosystems of evolutionary PDEs which have a common underlying gradient flow structure:in particular, we will focus onquasistationary models, which exhibithighly non convex Lyapunov functionals.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ambrosio, L., Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191246.
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, Clarendon Press, Oxford (2000).
L. Ambrosio, N Gigli and G. Savaré, Gradient flows.In metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag (2005).
C. Baiocchi, Discretization of evolution variational inequalities, Partial differential equations and the calculus of variations, Vol. I, F. Colombini, A. Marino, L. Modica and S. Spagnolo, Eds., Birkhäuser Boston, Boston, MA (1989) 59–92.
Balder, E.J., A general approach to lower semicontinuity and lower closure in optimal control theory. SIAM J. Control Optim. 22 (1984) 570598. CrossRef
Balder, E.J., An extension of Prohorov's theorem for transition probabilities with applications to infinite-dimensional lower closure problems. Rend. Circ. Mat. Palermo 34 (1985) 427447. CrossRef
E.J. Balder, Lectures on Young measure theory and its applications in economics. Rend. Istit. Mat. Univ. Trieste 31 (2000) (Suppl. 1), 1–69, Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1997).
Ball, J.M., A version of the fundamental theorem for Young measures, PDEs and continuum models of phase transitions (Nice 1988), Springer, Berlin. Lect. Notes Phys. 344 (1989) 207215. CrossRef
Bouchitté, G., Singular perturbations of variational problems arising from a two-phase transition model. Appl. Math. Optim. 21 (1990) 289314. CrossRef
Bressan, A., Cellina, A. and Colombo, G., Upper semicontinuous differential inclusions without convexity. Proc. Amer. Math. Soc. 106 (1989) 771775. CrossRef
H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contribution to Nonlinear Functional Analysis, in Proc. Sympos. Math. Res. Center, Univ. Wisconsin, Madison, 1971. Academic Press, New York (1971) 101–156.
H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Publishing Co., Amsterdam (1973), North-Holland Mathematics Studies, No. 5. Notas de Matemática (50).
H. Brézis, Analyse fonctionnelle - Théorie et applications. Masson, Paris (1983).
H. Brézis, On some degenerate nonlinear parabolic equations, Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 1, Chicago, Ill., 1968), Providence, R.I., Amer. Math. Soc. (1970) 28–38.
Cardinali, T., Colombo, G., Papalini, F. and Tosques, M., On a class of evolution equations without convexity. Nonlinear Anal. 28 (1997) 217234. CrossRef
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. Springer, Berlin-New York (1977).
Crandall, M.G. and Liggett, T.M., Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1971) 265298. CrossRef
Crandall, M.G. and Pazy, A., Semi-groups of nonlinear contractions and dissipative sets. J. Functional Anal. 3 (1969) 376418. CrossRef
E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications, C. Baiocchi and J.L. Lions, Eds., Masson (1993) 81–98.
De Giorgi, E., Marino, A. and 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) 180187.
C. Dellacherie and P.A. Meyer, Probabilities and potential. North-Holland Publishing Co., Amsterdam (1978).
Jordan, R., Kinderlehrer, D. and Otto, F., The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal. 29 (1998) 117 (electronic). CrossRef
T. Kato, Perturbation theory for linear operators. Springer, Berlin (1976).
Kōmura, Y., Nonlinear semi-groups in Hilbert space. J. Math. Soc. Japan 19 (1967) 493507. CrossRef
Kruger, A.Ja. and Mordukhovich, B.Sh., Extremal points and the Euler equation in nonsmooth optimization problems. Dokl. Akad. Nauk BSSR 24 (1980) 684687, 763.
Luckhaus, S., Solutions for the two-phase Stefan problem with the Gibbs-Thomson Law for the melting temperature. Euro. J. Appl. Math. 1 (1990) 101111. CrossRef
S. Luckhaus, The Stefan Problem with the Gibbs-Thomson law. Preprint No. 591 Università di Pisa (1991) 1–21.
Luckhaus, S., The Gibbs-Thompson relation within the gradient theory of phase transitions. Arch. Rational Mech. Anal. 107 (1989) 7183. CrossRef
Marino, A., Saccon, C. and Tosques, M., Curves of maximal slope and parabolic variational inequalities on nonconvex constraints. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989) 281330.
Mielke, A., Theil, F. and Levitas, V.I., A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162 (2002) 137177. CrossRef
Modica, L., Gradient theory of phase transitions and minimal interface criterion. Arch. Rational Mech. Anal. 98 (1986) 123142. CrossRef
Modica, L. and Mortola, S., Un esempio di ${\Gamma}$ -convergenza. Boll. Un. Mat. Ital. B 14 (1977) 285299.
Mordukhovich, B.Sh., Nonsmooth analysis with nonconvex generalized differentials and conjugate mappings. Dokl. Akad. Nauk BSSR 28 (1984) 976979.
Nochetto, R.H., Savaré, G. and Verdi, C., A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Comm. Pure Appl. Math. 53 (2000) 525589. 3.0.CO;2-M>CrossRef
Plotnikov, P.I. and Starovoitov, V.N., The Stefan problem with surface tension as the limit of a phase field model. Differential Equations 29 (1993) 395404.
R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton (1970).
R.T. Rockafellar and R.J.B. Wets, Variational analysis. Springer-Verlag, Berlin (1998).
R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces. Ann. Sc. Norm. Sup., Pisa 2 (2003) 395–431.
Rossi, R. and Savaré, G., Existence and approximation results for gradient flows. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (9) Mat. Appl. 15 (2004) 183196.
Rulla, J., Error analysis for implicit approximations to solutions to Cauchy problems. SIAM J. Numer. Anal. 33 (1996) 6887. CrossRef
Savaré, G., Weak solutions and maximal regularity for abstract evolution inequalities. Adv. Math. Sci. Appl. 6 (1996) 377418.
Savaré, G., Compactness properties for families of quasistationary solutions of some evolution equations. Trans. Amer. Math. Soc. 354 (2002) 37033722. CrossRef
Schätzle, R., The quasistationary phase field equations with Neumann boundary conditions. J. Differential Equations 162 (2000) 473503. CrossRef
L. Simon, Lectures on geometric measure theory, in Proc. Centre for Math. Anal., Australian Nat. Univ. 3 (1983).
M. Valadier, Young measures, Methods of nonconvex analysis (Varenna, 1989). Springer, Berlin (1990) 152–188.
A. Visintin, Differential models of hysteresis. Appl. Math. Sci. 111, Springer-Verlag, Berlin (1994).
A. Visintin, Models of phase transitions. Progress in Nonlinear Differential Equations and Their Applications 28, Birkhäuser, Boston (1996).
Visintin, A., Forward-backward parabolic equations and hysteresis. Calc. Var. Partial Differential Equations 15 (2002) 115132. CrossRef