to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the formWe assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f(x, s)s−1 → a ∈)0, ∞) as s →+∞; and (ii) f(x, s)s–1 is non decreasing as a function of s ≥ 0, a.e. x → ℝN.

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On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN

Variational methods for non-differentiable functionals and their applications to partial differential equations

CRITICAL POINT THEOREMS AND APPLICATIONS TO SOME NONLINEAR PROBLEMS WITH -STRONG” RESONANCE AT INFINITY + P. BARTOLO and V. BENCI Istituto di Matematica Applicata. Via Re David. X0-Bari. Ital) and Istituto di Analisi Matematica. Palazzo Ateneo. L’ia Scolai. 2-Bari. Italy (Received in revised form 20 December 1952)

Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity

In this paper we study the eigenvalue problem for the Schrödinger operator coupled with the electromagnetic field E,H. The case in which the electromagnetic field is given has been mainly considered ([1]–[3]). Here we do not assume that the electromagnetic field is assigned, then we have to study a system of equations whose unknowns are the wave function ψ = ψ(x, t) and the gauge potentials A = A(x, t), φ = φ(x, t) related to E,H. We want to investigate the case in which A and φ do not depend on the time t and ψ(x, t) = u(x)e, u real function and ω a real number In this situation we can assume A = 0 and we are reduced to study the existence of real numbers ω and real functions u, φ satisfying the system

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An eigenvalue problem for the Schrödinger-Maxwell equations

A variational principle of a minimax nature is developed and used to prove the existence of critical points for certain variational problems which are indefinite. The proofs are carried out directly in an infinite dimensional Hilbert space. Special cases of these problems previously had been tractable only by an elaborate finite dimensional approximation procedure. The main applications given here are to Hamiltonian systems of ordinary differential equations where the existence of time periodic solutions is established for several classes of Hamiltonians.

Critical point theorems for indefinite functionals

This paper is divided in two parts. In the first part we construct a model which describes solitary waves of the nonlinear Klein-Gordon equation interacting with the electromagnetic field. In the second part we study the electrostatic case. We prove the existence of infinitely many pairs (ψ, E), where ψ is a solitary wave for the nonlinear Klein-Gordon equation and E is the electric field related to ψ.

Solitary waves of the nonlinear klein-gordon equation coupled with the maxwell equations

We consider functionals which are not bounded from above or from below even modulo compact perturbations, and which exhibit certain symmetries with respect to the action of a compact Lie group. We develop a method which permits us to prove the existence of multiple critical points for such functionals. The proofs are carried out directly in an infinite dimensional Hilbert space, and they are based on minimax arguments. The applications given here are to Hamiltonian systems of ordinary differential equations where the existence of multiple time-periodic solutions is established for several classes of Hamiltonians. Symmetry properties of these Hamiltonians such as time translation invariancy or evenness are exploited. Introduction. Many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above. However some of these functionals, defined in an appropriate function space, are "semidefinite" in the sense that they are bounded from below (or from above) if perturbed with a weakly continuous functional. This paper deals with functionals which are not semidefinite. Usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space [7]. In [7] some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals. In many physical situations there are problems which have symmetries with respect to the action of some Lie group. In this case we expect the existence of many critical points; this has been established for semidefinite functionals (cf. e.g. [8 and 3] for even functionals, [9] for a Zp-action with p prime numbers [15, 16, 10 and 6] for an S'-action). In this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries. More precisely Received by the editors July 16, 1980 and, in revised form, June 26, 1981. 1980 Mathematics Subject Classification. Primary 58E05, 70H05; Secondary 34C25.

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Vieri Benci

Papers

Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity

An eigenvalue problem for the Schrödinger-Maxwell equations

Critical point theorems for indefinite functionals

Solitary waves of the nonlinear klein-gordon equation coupled with the maxwell equations

On critical point theory for indefinite functionals in the presence of symmetries