Abstract
We show that in viscous approximations of functionals defined on Finsler manifolds, it is possible to construct suitable sequences of critical points of these approximations satisfying the expected Morse index bounds as in Lazer–Solimini’s theory, together with the entropy condition of Michael Struwe.
Highlights
Theorem 1.1 Let (X, · X ) be a C2 Finsler manifold modelled on a Banach space E, and Y → X be a C2 Finsler–Hilbert manifold modelled on a Hilbert space H which we suppose locally Lipschitz embedded in X, and let F, G ∈ C2(X, R+) be two fixed functions
Let us emphasize that there is to our knowledge no method to prove directly Morse index estimates in this setting without reducing to the non-degenerate case, and the Fredholm hypothesis on the second derivative becomes at this point necessary as the only known way to perturb a function on a Finsler–Hilbert manifold to make it non-degenerate is to use the Sard–Smale theorem, for which this hypothesis is necessary
We say that F is a Fredholm map at x if D F(x) : Tx X → TF(x)Y is a Fredholm operator and we define the index of F at x, still denoted by Indx (F), by
Summary
As the multiplicity of the 0-eigenvalue) of the Fredholm operator ∇2 F(x) : Tx X → Tx X In this setting, it was subsequently proved by Lazer and Solimini [12] that it is possible to find a critical point x∗ ∈ K (F) (a priori different from x) such that the following index bound holds. In the framework of the viscosity method (see [15] for a general introduction on the subject), the function F does not satisfy the Palais–Smale condition (classical examples are given by minimal or Willmore surfaces), and one wishes to construct critical points of F by approaching F by a more coercive function for which we can apply the previous standard methods. (1) Palais–Smale condition For all σ > 0, the function Fσ : X → R+ satisfies the Palais– Smale condition at all positive level c > 0
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Calculus of Variations and Partial Differential Equations
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
