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Introduction to Morse theory. A new approach. (English) Zbl 0823.58008
Matzeu, Michele (ed.) et al., Topological nonlinear analysis: degree, singularity, and variations. Papers presented in part at the topological analysis workshop on degree, singularity, and variations, held in May 1993 in Frascati, Italy. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 15, 37-177 (1995).

Since the dawns of the branch of mathematics known as partial differential equations, the variational techniques played an important part, albeit controversial at the beginning.

The Morse theory first displayed the intimate interdependence [“analysis $↔$ topology”]. Until relatively recently this used to be a one way avenue: a good grasp of the analysis (or geometry) of the context yielded remarkable topological results (witness Bott’s results on the topology of symmetric spaces).

About three decades ago mathematicians began restoring the balance and use topology for the benefit of analysis. Their efforts paid off. However, for various reasons, there are very few instances where the full strength of Morse theory was actually used. This has to do with some constraints stemming form the infinite-dimensional background. We list some of them.

(a) Many functionals arising in nonlinear analysis are merely ${C}^{1}$ and not ${C}^{2}$ as the usual Morse theory requires.

(b) The infinite-dimensional Morse theory requires an added compactness assumption (like e.g. the Palais-Smale condition) which may not be fulfilled in concrete examples.

(c) The Morse theory works very well for non-degenerate functionals and almost always this condition is not met.

The survey under review represents one attempt to overcome these difficulties.

The author, which is a leading expert in the field, bases his approach on a Conley index type theory in which (c) is no longer a stringent requirement. As for the limitations (a) and (b), although there is no hope for a general theory, they can be dealt with on a case by case basis.

The first chapter of the survey is a presentation of the general techniques and principles of this new approach.

Section I.4 interprets the classical Morse theory in terms of the homological Conley index introduced in the Section I.3. The Morse inequalities for a nondegenerate functional follow immediately form these considerations.

The Section I.5 treats degenerate functionals. The theory works well for functionals $\left[f\in {C}^{1}\left(\overline{{\Omega }}\right)\right]$ (where ${\Omega }$ is an open subset of some Hilbert space $H$) such that for any (arbitrarily small) neighborhood $\nu$ of $\left\{\nabla f=0\right\}$ there exists a good Morse function $g\in {C}^{2}\left({\Omega }\right)\cap {C}^{1}\left(\overline{{\Omega }}\right)$ which agrees with $f$ outside $\nu$. Admittedly, this is a somewhat awkward condition to verify but the author supplements it with some explicit situations when this happens. For example, a ${C}^{2}$, Palais-Smale functional having only isolated critical points with Fredholm Hessians is admissible. This is precisely the type of functionals considered by Marino and Prodi in the 70’s. The index of an isolated (possibly degenerate) critical set is a Poincaré polynomial, not just a number. The author describes what kind of informations one can read from the coefficients of this polynomial. E.g. for the Marino-Prodi functionals each non-zero coefficient indicates the presence of a different critical point. The generalized Morse indices are related to the global behavior of the functional via the generalized Morse inequalities described in Theorem I.5.9.

In the last section of Chapter I the author studies the generalized Morse indices of the critical points obtained via Mountain Pass type theorems. The results are especially accurate for Marino-Prodi functionals.

Chapter II describes the favorite application of the Morse theory, i.e. the variational theory of geodesics in a Riemannian manifold. Although this describes only classical facts it helps the reader to understand the goal of the next Chapter, where the author initiates a program of extending the Morse theory of geodesics on a Riemannian manifold to geodesics on a Lorentz manifold. The Lorentz case is considerably more difficult and from the start one notices two problems.

(a) The energy functional is no longer bounded from below since the Lorentz metric is indefinite.

(b) The Hessian of a geodesic almost always has an infinite-dimensional negative subspace. This is just a reflection of the fact that on a Lorentz manifold the geodesic equation is hyperbolic.

The author concentrates on split type Lorentz spaces, i.e. Lorentz spaces diffeomorphic with a product $M×ℝ$ with Lorentz metric of the type $\left[{g}_{t}-\beta \left(x,t\right)d{t}^{2}\right]$, where ${g}_{t}$ is a time dependent Riemannian metric on $M$ and $\beta$ is an everywhere positive function.

The author describes a new functional $J$ whose critical points are the (Lorentz) geodesics. The advantage of $J$ over the usual energy functional is that the Hessians of the various critical points have finite Morse index which can be described as in the Riemannian case using the conjugate points along a geodesic. The Morse theory in the Riemannian case continue to hold, almost word by word, for the split type Lorentz manifolds satisfying some growth conditions on the metric.

The last chapter of the survey discusses various applications to semilinear elliptic equations (1) $-{\Delta }u+f\left(u\right)=0$ in ${\Omega }$, $u=0$ on $\partial {\Omega }$. The functionals arising in this case are of Marino-Prodi type. One first interesting result in this chapter is Theorem IV.1.4. It states that if $f$ satisfies some power like growth conditions and $\left[{f}^{\text{'}}\left(s\right){s}^{2}-f\left(s\right)s<0$, $\forall s\in ℝ\right]$ then the Morse index of a solution of (1) is bounded below by the number of nodal regions (where the solution $u$ has constant sign). A particularly nice result is Theorem IV.4.1. which deals with the positive solutions of $\left[-\epsilon {\Delta }u+u=f\left(u\right)$ in ${\Omega }$, $u=0$ on $\partial {\Omega }$ $\left(\epsilon >0\right)\right]$. $f$ satisfies a certain list of conditions. This theorem establishes (for $\epsilon$ sufficiently small) Morse type inequalities between the Morse indices of the positive solutions and the Poincaré polynomial of ${\Omega }$. It is therefore establishing in an explicit, quantifiable manner, the connection between the topology of ${\Omega }$ and the number of positive solutions. The proof of this result occupies the last two sections of the survey.

This work fills a gap in the present literature concerning variational techniques in nonlinear analysis. It presents in a friendly manner some basic topological techniques many analysts are not comfortable with. We welcome its apparition.

MSC:
 58E05 Abstract critical point theory 58E10 Applications of variational methods to geodesics 35J20 Second order elliptic equations, variational methods 58-02 Research monographs (global analysis) 53C50 Lorentz manifolds, manifolds with indefinite metrics