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Mathematical Sciences Colloquium by Augusto Visintin
Friday, Oct 20
11 a.m. - 11:50 a.m. Location: FN 2.102

Augusto Visintin

Trento, Italy

Compactness and Structural Stability of Nonlinear Flows

Parabolic equations are usually regarded as not representable by a minimization principle; however results of Brezis and Ekeland [1], Nayroles [3] and Fitzpatrick [4] provide evidence of the contrary. This can also be extended to the parabolic flow of pseudo-monotone operators.

Compactness and stability of the dependence of the solution on the data and the potential can be proved on the basis of De Giorgi's notion of Gamma-convergence [2]. Here by compactness we mean that any sequence of potentials has a Gamma-convergent subsequence. Stability means that, if the data converge and the potentials Gamma-converge, then the sequence of solutions converges to a solution, up to extracting a subsequence, [5]. This was already known for stationary problems, and is here extended to evolutionary problems via a novel notion of evolutionary Gamma-convergence.

These results can be extended in several directions, including doubly-nonlinear parabolic inclusions.

This research is surveyed in [6].

References
[1] H. Brezis, I. Ekeland: Un principe variationnel associe a certaines equations paraboliques. I. Le cas independant du temps, II. Le cas dependant du temps. C. R. Acad. Sci. Paris Ser. A-B 282 (1976) 971{974, and ibid. 1197{1198
[2] E. De Giorgi, T. Franzoni: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975) 842{850
[3] B. Nayroles: Deux theoremes de minimum pour certains systemes dissipatifs. C. R. Acad. Sci. Paris Ser. A-B 282 (1976) A1035{A1038
[4] S. Fitzpatrick: Representing monotone operators by convex functions. Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), 59{65, Proc. Centre Math. Anal. Austral. Nat. Univ., 20, Austral. Nat. Univ., Canberra, 1988
[5] A. Visintin: Variational formulation and structural stability of monotone equations. Calc. Var. Partial Di erential Equations 47 (2013), 273{317
[6] A. Visintin: On Fitzpatrick's theory and stability of flows. Rend. Lincei Mat. Appl. 27 (2016) 1{30

 

Coffee to be served in FN 2.102 at 10:30 AM.

 

Sponsored by the Department of Mathematical Sciences

Contact Info:
Viswanath Ramakrishna, 972-883-6873
Questions? Email me.

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