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Convexity and Generalized Convexity Methods for the Study of Hamilton-Jacobi Equations

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Generalized Convexity and Generalized Monotonicity

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 502))

Abstract

We present a digest of recent results about the first-order HamiltonJacobi equation. We use explicit formulas of the Hopf and Lax-Oleinik types, stressing the role of quasiconvex duality: here the usual Fenchel conjugacy is replaced with quasiconvex conjugacies known from some years and the usual inf-convolution is replaced by the sublevel convolution. The role of the full theory of variational convergences (epi-convergence and sublevel convergence) is put in light for the verification of initial conditions. We observe that duality methods and variational convergences are not limited to the case of finite-valued functions as in the classical approaches to Hamilton-Jacobi equations. This extension allows to deal with problems arising from various cases, such as optimal control problems, attainability or obstacle problems.

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Author information

Authors and Affiliations

  1. Faculté des Sciences, Laboratoire de Mathématiques Appliquées, Université de Pau, UPRES A 5033 Av. de l’Université, 64000, Pau, France

    Jean-Paul Penot

  2. Faculté des Sciences, Département, Université d’Avignon, de Mathématiques 33 rue Louis Pasteur, 84000, Avignon, France

    Michel Volle

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  1. Jean-Paul Penot
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  2. Michel Volle
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of the Aegean, 83200, Karlovassi/Samos, Greece

    Nicolas Hadjisavvas

  2. CODE and Department of Economics, Universitat Autonoma de Barcelona, 08193, Bellaterra, Spain

    Juan Enrique Martínez-Legaz

  3. Department of Mathematics Faculty of Sciences, Université de Pau, 64000, Pau, France

    Jean-Paul Penot

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Penot, JP., Volle, M. (2001). Convexity and Generalized Convexity Methods for the Study of Hamilton-Jacobi Equations. In: Hadjisavvas, N., Martínez-Legaz, J.E., Penot, JP. (eds) Generalized Convexity and Generalized Monotonicity. Lecture Notes in Economics and Mathematical Systems, vol 502. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56645-5_21

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  • DOI: https://doi.org/10.1007/978-3-642-56645-5_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41806-1

  • Online ISBN: 978-3-642-56645-5

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