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Thomas Bartsch - One of the best experts on this subject based on the ideXlab platform.

  • deformation theorems on non metrizable vector spaces and applications to Critical Point Theory
    Mathematische Nachrichten, 2006
    Co-Authors: Thomas Bartsch, Yanheng Ding
    Abstract:

    Let E be a Banach space and Φ : E ℝ a 1-functional. Let be a family of semi-norms on E which separates Points and generates a (possibly non-metrizable) topology on E weaker than the norm topology. This is a special case of a gage space, that is, a topological space where the topology is generated by a family of semi-metrics. We develop some Critical Point Theory for Φ : (E, ) ℝ. In particular, we prove deformation lemmas where the deformations are continuous with respect to . In applications this yields a gain in compactness when Φ does not satisfy the Palais–Smale condition because one can work with the weak topology. We also prove some foundational results on gage spaces. In particular, we introduce the concept of Lipschitz continuity in this setting and prove the existence of Lipschitz continuous partitions of unity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

  • Deformation theorems on non‐metrizable vector spaces and applications to Critical Point Theory
    Mathematische Nachrichten, 2006
    Co-Authors: Thomas Bartsch, Yanheng Ding
    Abstract:

    Let E be a Banach space and Φ : E ℝ a 1-functional. Let be a family of semi-norms on E which separates Points and generates a (possibly non-metrizable) topology on E weaker than the norm topology. This is a special case of a gage space, that is, a topological space where the topology is generated by a family of semi-metrics. We develop some Critical Point Theory for Φ : (E, ) ℝ. In particular, we prove deformation lemmas where the deformations are continuous with respect to . In applications this yields a gain in compactness when Φ does not satisfy the Palais–Smale condition because one can work with the weak topology. We also prove some foundational results on gage spaces. In particular, we introduce the concept of Lipschitz continuity in this setting and prove the existence of Lipschitz continuous partitions of unity. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

  • Critical Point Theory on partially ordered hilbert spaces
    Journal of Functional Analysis, 2001
    Co-Authors: Thomas Bartsch
    Abstract:

    We develop some abstract Critical Point Theory in order to prove that boundary value problems like the model problem[formula] on a bounded domain Ω⊂RN, 2supersolutions, Related results on the existence of sign-changing solutions hold for other classes of nonlinearities.

  • Critical Point Theory on Partially Ordered Hilbert Spaces
    Journal of Functional Analysis, 2001
    Co-Authors: Thomas Bartsch
    Abstract:

    We develop some abstract Critical Point Theory in order to prove that boundary value problems like the model problem[formula] on a bounded domain Ω⊂RN, 2

  • Critical Point Theory for Indefinite Functionals with Symmetries
    Journal of Functional Analysis, 1996
    Co-Authors: Thomas Bartsch, Mónica Clapp
    Abstract:

    LetXbe a Hilbert space andφ∈C1(X, R) be strongly indefinite. Assume in addition that a compact Lie groupGacts orthogonally onXand thatφis invariant. In order to find Critical Points ofφwe develop the limit relative category of Fournieret al. in the equivariant context. We use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem. In the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension. Also other conditions of the classical symmetric mountain pass theorem forG=Z/2 (due to Ambrosetti and Rabinowitz) can be weakened considerably. For example, we are able to deal with infinite-dimensional fixed Point spaces. The proofs consist of a direct reduction to a relative Borsuk–Ulam type theorem. This provides a new proof even for the classical mountain pass theorem. The abstract Critical Point theorems are applied to an elliptic system with Dirichlet boundary conditions. We only need a weak version of the usual superquadraticity condition. The linking theorem can be applied to asymptotically linear Hamiltonian systems which are symmetric with respect to a (generalized) symplectic group action.

Martin Schechter - One of the best experts on this subject based on the ideXlab platform.

  • Topics in Critical Point Theory
    2012
    Co-Authors: Kanishka Perera, Martin Schechter
    Abstract:

    Preface 1. Morse Theory 2. Linking 3. Applications to semilinear problems 4. Fucik spectrum 5. Jumping nonlinearities 6. Sandwich pairs Appendix: Sobolev spaces Bibliography Index.

  • minimax systems and Critical Point Theory
    2009
    Co-Authors: Martin Schechter
    Abstract:

    Preface.-.Critical Points of Functionals.-.Minimax Systems.-.Examples of Minimax Systems.-.Ordinary Differential Equations.-.The Method using Flows.-.Finding Linking Sets.-.Sandwich Pairs.-.Semilinear Problems.-.Superlinear Problems.-.Weak Linking.-.Resonance Problems.-.Rotationally Invariant Solutions.-.Semilinear Wave Equations.-.Type (II) Regions.-.Weak Sandwich Pairs.-.Multiple Solutions.-.Second Order Periodic Systems.-.Bibliography

  • Sandwich pairs in Critical Point Theory
    Transactions of the American Mathematical Society, 2008
    Co-Authors: Martin Schechter
    Abstract:

    Since the development of the calculus of variations there has been interest in finding Critical Points of functionals. This was intensified by the fact that for many equations arising in practice the solutions are Critical Points of functionals. If a functional G is semibounded, one can find a Palais-Smale (PS) sequence G(u k ) → a, G'(u k ) → 0. These sequences produce Critical Points if they have convergent subsequences (i.e., if G satisfies the PS condition). However, there is no clear method of finding Critical Points of functionals which are not semibounded. The concept of linking was developed to produce Palais-Smale (PS) sequences for C 1 functionals G that separate linking sets. In the present paper we discuss the situation in which one cannot find linking sets that separate the functional. We introduce a new class of subsets that accomplishes the same results under weaker conditions. We then provide criteria for determining such subsets. Examples and applications are given.

  • The Use of Cerami Sequences in Critical Point Theory
    Abstract and Applied Analysis, 2007
    Co-Authors: Martin Schechter
    Abstract:

    The concept of linking was developed to produce Palais-Smale (PS) sequences G ( u k ) → a , G ' ( u k ) → 0 for C 1 functionals G that separate linking sets. These sequences produce Critical Points if they have convergent subsequences (i.e., if G satisfies the PS condition). In the past, we have shown that PS sequences can be obtained even when linking does not exist. We now show that such situations produce more useful sequences. They not only produce PS sequences, but also Cerami sequences satisfying G ( u k ) → a , ( 1 + | | u k | | ) G ' ( u k ) → 0 as well. A Cerami sequence can produce a Critical Point even when a PS sequence does not. In this situation, it is no longer necessary to show that G satisfies the PS condition, but only that it satisfies the easier Cerami condition (i.e., that Cerami sequences have convergent subsequences). We provide examples and applications. We also give generalizations to situations when the separating criterion is violated.

  • Critical Point Theory and its applications
    2006
    Co-Authors: Wenming Zou, Martin Schechter
    Abstract:

    Preliminaries.- Functionals Bounded Below.- Even Functionals.- Linking and Homoclinic Type Solutions.- Double Linking Theorems.- Superlinear Problems.- Systems with Hamiltonian Potentials.- Linking and Elliptic Systems.- Sign-Changing Solutions.- Cohomology Groups.

Michel Willem - One of the best experts on this subject based on the ideXlab platform.

Marco Degiovanni - One of the best experts on this subject based on the ideXlab platform.

  • On topological and metric Critical Point Theory
    Journal of Fixed Point Theory and Applications, 2009
    Co-Authors: Marco Degiovanni
    Abstract:

    Starting from the concept of Morse Critical Point, introduced in [19], we propose a possible approach to Critical Point Theory for continuous functionals defined on topological spaces, which includes some classical results, also in an infinite-dimensional setting.

  • A Survey on Nonsmooth Critical Point Theory and Applications
    Nonconvex Optimization and Its Applications, 2001
    Co-Authors: Marco Degiovanni
    Abstract:

    In the recent years, new advances have been obtained in Critical Point Theory for nonsmooth functionals and in applications to nonlinear differential equations. Here we provide a survey on some of such progresses.

  • Subdifferential Calculus and Nonsmooth Critical Point Theory
    SIAM Journal on Optimization, 2000
    Co-Authors: Ines Campa, Marco Degiovanni
    Abstract:

    A general Critical Point Theory for continuous functions defined on metric spaces has been recently developed. In this paper a new subdifferential, related to that Theory, is introduced. In particular, results on the subdifferential of a sum are proved. An example of application to PDEs is sketched. Detailed applications to PDEs are developed in separate papers.

  • Perturbations of Critical Values in Nonsmooth Critical Point Theory
    Serdica. Mathematical Journal, 1996
    Co-Authors: Marco Degiovanni, Sergio Lancelotti
    Abstract:

    The perturbation of Critical values for continuous functionals is stud- ied. An application to eigenvalue problems for variational inequalities is provided.

  • Nonsmooth Critical Point Theory and quasilinear elliptic equations
    Topological Methods in Differential Equations and Inclusions, 1995
    Co-Authors: Annamaria Canino, Marco Degiovanni
    Abstract:

    These lectures are devoted to a generalized Critical Point Theory for nonsmooth functionals and to existence of multiple solutions for quasilinear elliptic equations. If f is a continuous function defined on a metric space, we define the weak slope |df|(u), an extended notion of norm of the Frechet derivative. Generalized notions of Critical Point and Palais-Smale condition are accordingly introduced. The Deformation Theorem and the NonCritical Interval Theorem are proved in this setting. The case in which f is invariant under the action of a compact Lie group is also considered. Mountain pass theorems for continuous functionals are proved. Estimates of the number of Critical Points of f by means of the relative category are provided. A partial extension of these techniques to lower semicontinuous functionals is outlined. The second part is mainly concerned with functionals of the Calculus of Variations depending quadratically on the gradient of the function. Such functionals are naturally continuous, but not locally Lipschitz continuous on H 0 1 . When f is even and suitable qualitative conditions are satisfied, we prove the existence of infinitely many solutions for the associated Euler equation. The regularity of such solutions is also studied.

Mónica Clapp - One of the best experts on this subject based on the ideXlab platform.

  • Critical Point Theory of symmetric functions and closed geodesics
    Differential Geometry and its Applications, 1996
    Co-Authors: Mónica Clapp, Dieter Puppe
    Abstract:

    Abstract We develop a version of equivariant Critical Point Theory particularly adapted to finding closed geodesics by variational methods and use it to improve the known lower bounds for the number of “short” closed geodesics on some closed Riemannian manifolds.

  • Critical Point Theory for Indefinite Functionals with Symmetries
    Journal of Functional Analysis, 1996
    Co-Authors: Thomas Bartsch, Mónica Clapp
    Abstract:

    LetXbe a Hilbert space andφ∈C1(X, R) be strongly indefinite. Assume in addition that a compact Lie groupGacts orthogonally onXand thatφis invariant. In order to find Critical Points ofφwe develop the limit relative category of Fournieret al. in the equivariant context. We use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem. In the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension. Also other conditions of the classical symmetric mountain pass theorem forG=Z/2 (due to Ambrosetti and Rabinowitz) can be weakened considerably. For example, we are able to deal with infinite-dimensional fixed Point spaces. The proofs consist of a direct reduction to a relative Borsuk–Ulam type theorem. This provides a new proof even for the classical mountain pass theorem. The abstract Critical Point theorems are applied to an elliptic system with Dirichlet boundary conditions. We only need a weak version of the usual superquadraticity condition. The linking theorem can be applied to asymptotically linear Hamiltonian systems which are symmetric with respect to a (generalized) symplectic group action.

  • Critical Point Theory for perturbations of symmetric functionals
    Commentarii Mathematici Helvetici, 1996
    Co-Authors: Mónica Clapp
    Abstract:

    Functionals which are invariant under the action of a compact transformation groupG often have many Critical values. Here we consider functionals which are notG-invariant and give conditions for them to have infinitely many Critical values; including a mountain pass theorem. We apply it to prove the existence of infinitely many solutions of a nonlinear Dirichlet problem with perturbedG-symmetries.