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The Topology of Critical Sets of Some Ordinary Differential Operators

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Book cover Contributions to Nonlinear Analysis

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 66))

Abstract

We survey recent work of Burghelea, Malta and both authors on the topology of critical sets of nonlinear ordinary differential operators. For a generic nonlinearity f, the critical set of the first order nonlinear operator F 1(u)(t) = u′(t) + f(u(t)) acting on the Sobolev space H 1p of periodic functions is either empty or ambient diffeomorphic to a hyperplane. For the second order operator F 2(u)(t) = −u″(t) + f(u(t)) on H 2D (Dirichlet boundary conditions), the critical set is ambient diffeomorphic to a union of isolated parallel hyperplanes. For second order operators on H 2p , the critical set is not a Hilbert manifold but is still contractible and admits a normal form. The third order case is topologically far more complicated.

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Author information

Authors and Affiliations

  1. Departamento de Matemática, PUC-Rio, R. Marquês de S. Vicente 225, Rio de Janeiro, RJ 22453-900, Brazil

    Nicolau C. Saldanha & Carlos Tomei

Authors
  1. Nicolau C. Saldanha
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  2. Carlos Tomei
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Editor information

Editors and Affiliations

  1. Laboratoire Jacques-Louis Lions B.C. 187, Université Pierre et Marie Curie, 4, place Jussieu, 75252, Paris Cedex 05, France

    Thierry Cazenave

  2. Department of Mathematical Sciences, University of Nevada, Las Vegas, NV, 89154-4020, USA

    David Costa

  3. Instituto de Matemática, UNICAMP - IMECC, Caixa Postal: 6065, 13083-859, Campinas, SP, Brasil

    Orlando Lopes

  4. Departamento de Ingenieria Matemática Facultad de Ciencias Fisicas y Matemáticas, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile

    Raúl Manásevich

  5. Mathematics Department, University of Wisconsin-Madison, 480 Lincoln Dr, Madison, WI, 53706-1388, USA

    Paul Rabinowitz

  6. Dipartimento di Matematica, Università degli Studi, Via Saldini 50, 20133, Milano, Italy

    Bernhard Ruf

  7. Departamento de Matemática, PUC Rio, Rua Marquês de São Vicente, 225 EdifÍcio Cardeal Leme, Gávea - Rio de Janeiro, 22453-900, Brasil

    Carlos Tomei

Additional information

Dedicated to Djairo Figueiredo, with affection and admiration.

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Saldanha, N.C., Tomei, C. (2005). The Topology of Critical Sets of Some Ordinary Differential Operators. In: Cazenave, T., et al. Contributions to Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 66. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7401-2_33

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