Abstract.
We verify – after appropriate modifications – an old conjecture of Brezis-Ekeland ([3], [4]) concerning the feasibility of a global variational approach to the problems of existence and uniqueness of gradient flows for convex energy functionals. Our approach is based on a concept of ‘‘self-duality’’ inherent in many parabolic evolution equations, and motivated by Bolza-type problems in the classical calculus of variations. The modified principle allows to identify the extremal value –which was the missing ingredient in [3]– and so it can now be used to give variational proofs for the existence and uniqueness of solutions for the heat equation (of course) but also for quasi-linear parabolic equations, porous media, fast diffusion and more general dissipative evolution equations.
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References
Auchmuty, G.: Saddle points and existence-uniqueness for evolution equations. Diff. Integral Eqs. 6, 1161–1171 (1993)
Brezis, H.: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North Holland, Amsterdam-London, 1973
Brezis, H., Ekeland, I.: Un principe variationnel associé à certaines equations paraboliques. Le cas independant du temps. C.R. Acad. Sci. Paris Sér. A 282, 971–974 (1976)
Brezis, H., Ekeland, I.: Un principe variationnel associé à certaines equations paraboliques. Le cas dependant du temps. C.R. Acad. Sci. Paris Sér. A 282, 1197–1198 (1976)
Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Springer- Verlag, New York, 1977
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, Amer. Math. Soc., Providence, 1998
Ghoussoub, N., McCann, R.: A least action principle for steepest descent in a non-convex landscape. January 2004, To appear in Contemporary Math.
Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Eqs. 26, 101–174 (2001)
Rockafellar, R.T.: Existence and duality theorems for convex problems of Bolza. Trans. Am. Math. Soc. 159, 1–40 (1971)
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Both authors were partially supported by a grant from the Natural Science and Engineering Research Council of Canada.
This paper is part of this author’s Master’s thesis under the supervision of the first named author.
Revised version: 31 March 2004
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Ghoussoub, N., Tzou, L. A variational principle for gradient flows. Math. Ann. 330, 519–549 (2004). https://doi.org/10.1007/s00208-004-0558-6
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DOI: https://doi.org/10.1007/s00208-004-0558-6