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### Compactness and Structural Stability of Nonlinear Flows

##### Prof. Augusto Visintin

Sala conferenze IMATI-CNR, Pavia - Tuesday, October 3, 2017 h.15:00

**Abstract. **

After the seminal works [1]–[3] of Brezis, Ekeland, Nayroles and Fitzpatrick, maximal monotone operators α : V → P(V′) (V being a Banach space) and flows of the form

du/dt + α(u) ∋ h in V′, a.e. in ]0, T [, u(0) = u_{0}

can be formulated as a minimization principle, even if α is not a subdifferential.

On this basis, De Giorgi’s notion of Γ-convergence may be applied to the analysis of monotone inclusions. Compactness and structural stability of the Cauchy problem are then studied, with respect to arbitrary variations not only of the datum h ∈ L

^{2}(0,T;V′), but also of the operator α. This rests upon the use of an exotic nonlinear topology of weak type, and on the novel notion of

*evolutionary *Γ

*-convergence*, [4]–[6].

The operator α may also be assumed to be a multivalued pseudo-monotone operator, e.g.:

α(u) = −∇ · γ(u, ∇u) ∀u ∈ W_{0}^{1,p}(Ω),

with γ lower semicontinuous (as a multivalued operator) w.r.t. the first argument, and maximal monotone w.r.t. the second one.

These results can be extended in several directions, and can be applied to nonlinear either stationary of evolutionary PDEs, including doubly-nonlinear inclusions of the form

D_{t}∂φ(u) + α(u) ∋ h or α(D_{t}u) + ∂φ(u) ∋ h,

with α as above and φ convex and lower semicontinuous.

This research is surveyed in [6].

References.

[1] H. Brezis, I. Ekeland: Un principe variationnel associé à certaines équations paraboliques. I. Le cas indépendant du temps, II. Le cas dépendant du temps. C. R. Acad. Sci. Paris Sér. A-B 282 (1976) 971–974, and ibid. 1197–1198

[2] B. Nayroles: Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Sér. A-B 282 (1976) A1035–A1038

[3] 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

[4] A. Visintin: Extension of the Brezis-Ekeland-Nayroles principle to monotone operators. Adv. Math. Sci. Appl. 18 (2008) 633–650

[5] A. Visintin: Variational formulation and structural stability of monotone equations. Calc. Var. Partial Differential Equations 47 (2013), 273–317

[6] A. Visintin: On Fitzpatrick’s theory and stability of flows. Rend. Lincei Mat. Appl. 27 (2016) 1–30

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