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On the Structural Properties of Nonlinear Flows

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Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs

Part of the book series: Springer INdAM Series ((SINDAMS,volume 22))

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Abstract

This work deals with the structural stability of quasilinear first-order flows w.r.t. arbitrary perturbations not only of data but also of operators. This rests upon a variational formulation based on works of Brezis, Ekeland, Nayroles and Fitzpatrick, and on the use of evolutionary Γ-convergence w.r.t. a nonlinear topology of weak type. This approach is extended to flows of a class of nonmonotone operators. A theory in progress is outlined, and is also used to prove the structural compactness and stability of doubly-nonlinear parabolic flows of the form

$$\displaystyle\begin{array}{rcl} \alpha (D_{t}u) + \partial \gamma (u) \ni h,& & {}\\ \end{array}$$

α being a maximal monotone operator, and γ a lower semicontinuous convex function on a Hilbert space.

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Notes

  1. 1.

    This author learned of this paper only recently from Ulisse Stefanelli.

  2. 2.

    The proof was included in [21]. As those proceedings may not be easily available to the Reader, the argument was displayed also in [47].

  3. 3.

    We assume this in consideration of the application of the next section. A reader interested just in evolutionary Γ-convergence might go through the present section assuming that μ is the Lebesgue measure and that τ is the weak topology.

  4. 4.

    We denote the strong, weak, and weak star convergence respectively by →, ⇀, ⇀ ​​​​​​∗  .

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Acknowledgements

This paper is dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. The author is a member of GNAMPA of INdAM.

This research was partially supported by a MIUR-PRIN 2015 grant for the project “Calcolo delle Variazioni” (Protocollo 2015PA5MP7-004).

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Authors and Affiliations

  1. Dipartimento di Matematica dell’Università degli Studi di Trento, via Sommarive 14, 38123, Povo di Trento, Italy

    Augusto Visintin

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  1. Augusto Visintin
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Correspondence to Augusto Visintin .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Pierluigi Colli

  2. Dipartimento di Matematica, Università di Bologna , Bologna, Italy

    Angelo Favini

  3. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Elisabetta Rocca

  4. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giulio Schimperna

  5. für Angewandte Analysis und Stochastik (WIAS), Weierstraß-Institut , Berlin, Germany

    Jürgen Sprekels

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Visintin, A. (2017). On the Structural Properties of Nonlinear Flows. In: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (eds) Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-64489-9_21

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