Critical Point Theory

This book is divided into two parts. Part I is a modern introduction to the very classical theory of submanifold geometry. We go beyond the classical theory in at least one important respect; we study submanifolds of Hilbert space as well as of Euclidean spaces. Part II is devoted to critical point theory, and here again the theory is developed in the setting of Hilbert manifolds. The two parts are inter-related through the Morse Index Theorem, that is, the fact that the structure of the set of critical points of the distance function from a point to a submanifold can be described completely in terms of the local geometric invariants of the submanifold.

In this article, we consider a class of nonlinear Dirichlet problems driven by a Leray-Lions type operator with variable exponent. The main result establishes an existence property by means of nonvariational arguments, that is, nonlinear monotone operator theory and approximation method. Under some natural conditions, we show that a weak limit of approximate solutions is a solution of the given quasilinear elliptic partial differential equation involving variable exponent.

Working in a given conformal class, we prove existence of constant Q-curvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a nth-order nonlinear elliptic differential (or integral) equation with variational structure, where n is the dimension of the manifold. Since the corresponding Euler functional is in general unbounded from above and below, we use critical point theory, jointly with a compactness result for the above equation.

We study periodic flexings of a floating beam via critical point theory. The beam is described by the nonlinear equation with free-end boundary conditions described in the Introduction. We give a theorem of existence of two nontrivial solutions and a theorem of existence of four nontrivial solutions, depending on the position of a suitable parameter with respect to the eigenvalues of the linear part.

We combine topological and geometric methods to construct a multi-resolution representation for a function over a two-dimensional domain. In a preprocessing stage, we create the Morse-Smale complex of the function and progressively simplify its topology by canceling pairs of critical points. Based on a simple notion of dependency among these cancellations we construct a hierarchical data structure supporting traversal and reconstruction operations similarly to traditional geometrybased representations. We use this data structure to extract topologically valid approximations that satisfy error bounds provided at run-time.

We establish some framework so that the generalized Conley index can be easily used to study the multiple solution problem of semilinear elliptic boundary value problems. Both the parabolic flow and the gradient flow are used. Some examples are given to compare our approach here with other well-known methods. Our abstract results with parabolic flows may have applications to parabolic problems as well.

Dedicated to Andrzej Granas, with gratitude Abstract. In the framework of critical point theory for continuous func- tionals defined on metric spaces, we give a new, simpler proof of the so- called Second Deformation Lemma, a basic tool of Morse theory.

A class of extended real valued functionals, already considered for evolution problems, is studied. The set where the functional is finite is proved to be an absolute neighborhood extensor. Applications to critical point theory, involving Ljusternik-Schnirelman category and cohomological index, are shown. The stability under Γ-convergence of the homotopical type of the sublevels of the functional is also treated.

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all R. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of R. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).

In 1963, Palais and Smale have introduced a compactness condition, namely Condition (C), on real functions of class C 1 defined on a Riemannian manifold modelled upon a Hilbert space, in order to extend Morse theory to this frame and study nonlinear partial differential equations. This condition and some of its variants have been essential in the development of critical point theory on Banach spaces or Banach manifolds, and are referred as Palais-Smale-type conditions. The paper describes their evolution.

We study the existence of multiple positive solutions for a superlinear elliptic PDE with a sign-changing weight. Our approach is variational and relies on classical critical point theory on smooth manifolds. A special care is paid to the localization of minimax critical points.

We prove a critical-point result which provides conditions for the existence of infinitely many critical points of a strongly indefinite functional with perturbed symmetries. Then we apply this result to obtain infinitely many solutions of non-symmetric super-quadratic noncooperative elliptic systems, allowing some supercritical growth.

To gain a better insight into the effects of eddy structure and their thermal imprint on the impinging surface, we applied several methods to identify coherent structures in two arrangements of multiple jets. High-resolution particle imaging velocimetry measurements of the instantaneous fluid velocity have been performed to provide data for structure identification. Several methods based on critical point theory, i.e., vorticity magnitude, kinematic vorticity number, and second invariant of the velocity gradient tensor provided similar and generally useful information. However, in the flow considered they all appear inferior as compared with the proper orthogonal decomposition ͑POD͒. Despite lacking in conspicuous high energy modes, the flow showed to be well suited for POD, which provided clear identification of the near wall dominating vortical structure. Downloaded 27 Aug 2010 to 131.180.130.114. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 055105-2 Geers, Tummers, and Hanjalić Phys. Fluids 17, 055105 ͑2005͒ Downloaded 27 Aug 2010 to 131.180.130.114. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 055105-13 PIV-based identification of coherent structures Phys. Fluids 17, 055105 ͑2005͒

We combine topological and geometric methods to construct a multi-resolution data structure for functions over two-dimensional domains. Starting with the Morse-Smale complex, we construct a topological hierarchy by progressively canceling critical points in pairs. Concurrently, we create a geometric hierarchy by adapting the geometry to the changes in topology. The data structure supports mesh traversal operations similarly to traditional multiresolution representations.

We prove existence results for complex-valued solutions for a semilinear Schrödinger equation with critical growth under the perturbation of an external electromagnetic field. Solutions are found via an abstract perturbation result in critical point theory, developed in [1, 2, 5].

the particle velocity data is generated from a numerical simulation; therefore, the velocity data is only available at Several techniques for the numerical integration of particle paths in steady and unsteady vector (velocity) fields are analyzed. Most some set of discrete times, t n , for n ϭ 0, 1, 2, ..., N. The of the analysis applies to unsteady vector fields, however, some particle paths may be calculated in a postprocessing or results apply to steady vector field integration. Multistep, concurrent (coprocessing) manner. For postprocessing, the multistage, and some hybrid schemes are considered. It is shown time planes of velocity data are stored at some set number that due to initialization errors, many unsteady particle path integraof iterations typically determined by the amount of disk tion schemes are limited to third-order accuracy in time. Multistage schemes require at least three times more internal data storage storage available. For concurrent processing, the vector than multistep schemes of equal order. However, for timesteps field can be updated every iteration of the flow algorithm within the stability bounds, multistage schemes are generally more or after some set number of iterations. In either case, the accurate. A linearized analysis shows that the stability of these velocity field, u, is only available at some set of discrete integration algorithms are determined by the eigenvalues of the instances in time. A consequence of the temporally discrete local velocity tensor. Thus, the accuracy and stability of the methods are interpreted with concepts typically used in critical point theory. nature of the velocity data is that the timestep cannot be set This paper shows how integration schemes can lead to erroneous by the integration algorithm. This is unlike instantaneous classification of critical points when the timestep is finite and fixed. streamline integration (or steady velocity fields) for which For steady velocity fields, we demonstrate that timesteps outside the timestep can be adaptively varied to account for rapid of the relative stability region can lead to similar integration errors.

Communicated by R. P. Gilbert SUMMARY The aim of this paper is to establish the in uence of a non-symmetric perturbation for a symmetric hemivariational eigenvalue inequality with constraints. The original problem was studied by Goeleven et al. (Math. Methods Appl. Sci. 1997; 20:548) who deduced the existence of inÿnitely many solutions for the symmetric case. In this paper it is shown that the number of solutions of the perturbed problem becomes larger and larger if the perturbation tends to zero with respect to a natural topology. The approach relies on topological methods in non-smooth critical point theory leading to this new multiplicity information.

SEVERAL years ago the author, and independently Smale, generalized the Morse theory of critical points to cover certain functions on hilbert manifolds [5,6 and 91. Shortly thereafter J. Schwartz showed how the same techniques allowed one also to extend the Lusternik-Schnirelman theory of critical points to functions on hilbert manifolds [7]. Now while Morse theory, by its nature, is more or less naturally restricted to hilbert manifolds, the natural context for Lusternik-Schnirelman theory is a smooth real valued function on a Banach manifold. In this paper we shall extend Schwartz's results to cover this more general situation. It is not however simply a desire for generality in an abstract theorem that prompted the present work. The motivation for considering critical point theory in the context of infinite dimensional manifolds comes from the immediate applicability of results to proving existence theorems in the Calculus of Variations. As long as the integrands one considers are of a basically quadratic nature, hilbert manifolds of maps belonging to some Sobolev space Lf are the natural manifolds in which to formulate the problems. However for more general integrands it becomes necessary to use Banach manifolds, for example manifolds of maps belonging to one of the more general Sobolev spaces Lkp. Applications in this direction will be found in a forthcoming paper of Browder [2].

An application of ¢ ¤ £ scalar interpolation for 2D vector field topology visualization is presented. Powell-Sabin and Nielson interpolants are considered which both make use of Nielson's Minimum Norm Network for the precomputation of the derivatives in our implementation. A comparison of their results to the commonly used linear interpolant underlines their significant improvement of singularities location and topological skeleton depiction. Evalution is based upon the processing of polynomial vector fields with known topology containing higher order singularities.

We establish the existence of nontrivial solutions to sys- tems of singular Poisson equations in unbounded domains, under some invariance conditions and singular subcritical growth. The proofs rely on a concentration-compactness argument and on a generalized linking theorem due to Krysewski and Szulkin.

In this paper we prove the existence of infinitely many nontrivial solutions of the system ∆u = u, ∆v = v, with nonlinear coupling at the smooth boundary of a bounded domain of R N. The proof, under suitable assumptions on the Hamiltonian, is based on variational arguments and on the Fountain Theorem of the critical point theory.

In this paper we survey some recent results on the existence and multiplicity of radial solutions for Neumann problems in a ball and in an annular domain, associated to pendulum-like perturbations of mean curvature operators in Euclidean and Minkowski spaces and of the p-Laplacian operator. Our approach relies on the Leray-Schauder degree, upper and lower solutions method, and critical point theory.

We combine topological and geometric methods to construct a multi-resolution representation for a function over a two-dimensional domain. In a preprocessing stage, we create the Morse-Smale complex of the function and progressively simplify its topology by canceling pairs of critical points. Based on a simple notion of dependency among these cancellations we construct a hierarchical data structure supporting traversal and reconstruction operations similarly to traditional geometrybased representations. We use this data structure to extract topologically valid approximations that satisfy error bounds provided at run-time.

We define a Grushin-type operator with a variable exponent and establish existence results for an equation involving such an operator in a suitable function space. The tools used in proving our existence result rely on the critical point theory combined with adequate variational techniques.

An application of ¢ ¤ £ scalar interpolation for 2D vector field topology visualization is presented. Powell-Sabin and Nielson interpolants are considered which both make use of Nielson's Minimum Norm Network for the precomputation of the derivatives in our implementation. A comparison of their results to the commonly used linear interpolant underlines their significant improvement of singularities location and topological skeleton depiction. Evalution is based upon the processing of polynomial vector fields with known topology containing higher order singularities.