Riesz transform and integration by parts formulas for random variables

We use integration by parts formulas to give estimates for the L^p norm of the Riesz transform. This is motivated by the representation formula for conditional expectations of functionals on the Wiener space already given in Malliavin and Thalmaier (2006) [13]. As a consequence, we obtain regularity and estimates for the density of non-degenerated functionals on the Wiener space. We also give a semi-distance which characterizes the convergence to the boundary of the set of the strict positivity points for the density.     

A nondifferentiable extension of a theorem of Pucci and Serrin and applications

We study the multiplicity of critical points for functionals which are only differentiable along some directions. We extend to this class of functionals the three critical point theorem of Pucci and Serrin and we apply it to a one-parameter family of functionals Jλ, λ∈I⊂R. Under suitable assumptions, we locate an open subinterval of values λ in I for which Jλ possesses at least three critical points. Applications to quasilinear boundary value problems are also given.     

Bifurcation of critical points for continuous families of C 2 functionals of Fredholm type

Given a continuous family of C ² functionals of Fredholm type, we show that the nonvanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points in the parameter space and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization.     

Multiple critical points for non-differentiable parametrized functionals and applications to differential inclusions

In this paper we deal with a class of non-differentiable functionals defined on a real reflexive Banach space X and depending on a real parameter of the form ${\mathcal{E}_\lambda(u)=L(u)-(J_1\circ T)(u)-\lambda (J_2\circ S)(u)}$ , where ${L:X \rightarrow \mathbb R}$ is a sequentially weakly lower semicontinuous C ¹ functional, ${J_1:Y\rightarrow\mathbb R, J_2:Z\rightarrow \mathbb R}$ (Y, Z Banach spaces) are two locally Lipschitz functionals, T : X → Y, S : X → Z are linear and compact operators and λ > 0 is a real parameter. We prove that this kind of functionals posses at least three nonsmooth critical points for each λ > 0 and there exists λ* > 0 such that the functional ${\mathcal{E}_{\lambda^\ast}}$ possesses at least four nonsmooth critical points. As an application, we study a nonhomogeneous differential inclusion involving the p(x)-Laplace operator whose weak solutions are exactly the nonsmooth critical points of some “energy functional” which satisfies the conditions required in our main result.     

Another version of the proximal point algorithm in a Banach space

In this paper, we study some non-traditional schemes of proximal point algorithm for nonsmooth convex functionals in a Banach space. The proximal approximations to their minimal points and/or their minimal values are considered separately for unconstrained and constrained minimization problems on convex closed sets. For the latter we use proximal point algorithms with the metric projection operators and first establish the estimates of the convergence rate with respect to functionals. We also investigate the perturbed projection proximal point algorithms and prove their stability. Some results concerning the classical proximal point method for minimization problems in a Banach space is also presented in this paper.     

Spectral Flow,Maslov Index and Bifurcation of Semi-Riemannian Geodesics

We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals imply that each nondegenerate and nonnull conjugate (or P-focal)point along a semi-Riemannian geodesic is a bifurcation point.In particular, the semi-Riemannian exponential map is notinjective in any neighborhood of a nondegenerate conjugate point,extending a classical Riemannian result originally due to Morse and Littauer.     

Exit time moments, boundary value problems, and the geometry of domains in Euclidean space

Let X _t be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using the moments of the exit time for particles driven by X _t , as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all critical points are maxima. Received: 23 January 1997 / Revised version: 21 January 1998     

Bahri-Coron type theorem for the scalar curvature problem on high dimensional spheres

We consider the existence and multiplicity results for the prescribed scalar curvature problem on the standard spheres of high dimension n ?? 7. Given a C ² positive function K, using the theory of critical points at infinity, we prove an existence result as Bahri-Coron theorem. Our case is a generalization of Li (J Differ Equ 120:319?C410, 1995). Indeed, here the function K is flat near some critical points as in Li (J Differ Equ 120:319?C410, 1995) and it can have some nondegenerate critical points with ?? K ?? 0. Furthermore, using some topological arguments, we prove another kind of result.     

Generalized Brownian functionals and the Feynman integral

To obtain a sufficiently rich class of nonlinear functionals of white noise, resp. the Wiener process, we study riggings of the L² space with the white noise measure. Particular examples are local functionals such as e.g. the ‘square of white noise’ and its exponential with applications in the theory of Feynman Integral.     

Existence results for solutions to nonlinear Dirac equations on compact spin manifolds

We consider super-linear and sub-linear nonlinear Dirac equations on compact spin manifolds. Their solutions are obtained as critical points of certain strongly indefinite functionals on a Hilbert space. For both cases, we establish existence results via Galerkin type approximations and linking arguments. For a particular case of odd nonlinearities, we prove the existence of infinitely many solutions.     

Critical homoclinic orbits lead to snap-back repellers

When nondegenerate homoclinic orbits to an expanding fixed point of a map f:X→X,X⊆Rⁿ, exist, the point is called a snap-back repeller. It is known that the relevance of a snap-back repeller (in its original definition) is due to the fact that it implies the existence of an invariant set on which the map is chaotic. However, when does the first homoclinic orbit appear? When can other homoclinic explosions, i.e., appearance of infinitely many new homoclinic orbits, occur? As noticed by many authors, these problems are still open. In this work we characterize these bifurcations, for any kind of map, smooth or piecewise smooth, continuous or discontinuous, defined in a bounded or unbounded closed set. We define a noncritical homoclinic orbit and a homoclinic orbit of an expanding fixed point is structurally stable iff it is noncritical. That is, only critical homoclinic orbits are responsible for the homoclinic explosions. The possible kinds of critical homoclinic orbits will be also investigated, as well as their dynamic role.     

Pseudospectra, critical points and multiple eigenvalues of matrix polynomials

We develop a general framework for perturbation analysis of matrix polynomials. More specifically, we show that the normed linear space L_m(C^n×n) of n-by-n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in L_m(C^n×n). We present a family of natural norms on the space L_m(C^n×n) and show that the norms on the spaces C^m+1 and C^n×n play a crucial role in the perturbation analysis of matrix polynomials. We define pseudospectra of matrix polynomials in the general framework of the normed space L_m(C^n×n) and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases. We analyze various properties of pseudospectra in the unified framework of the normed space L_m(C^n×n). We analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial. We show that common boundary points of components of pseudospectra of matrix polynomials are critical points. As a consequence, we show that a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials.     

Morse theory for some lower-C 2 functions in finite dimension

Using inf-regularization methods, we prove that Morse inequalities hold for some lower-C ² functions. For this purpose, we first recall some properties of the class of lower-C ² functions and of their Moreau-Yosida approximations. Then, we establish, under some qualification conditions on the critical points, that it is possible to define a Morse index for a lower-C ² functionf. This index is preserved by the Moreau-Yosida approximation process. We prove in particular that the Moreau-Yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space. In a last step, Morse inequalities are written for Moreau-Yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower-C ² functions.     

A computable error bound for matrix functionals

Many problems in applied mathematics require the evaluation of matrix functionals of the form F(A):=u^Tf(A)u, where A is a large symmetric matrix and u is a vector. Golub and collaborators have described how approximations of such functionals can be computed inexpensively by using the Lanczos algorithm. The present note shows that error bounds for these approximations can be computed essentially for free when bounds for derivatives of f on an interval containing the spectrum of A are available.     

Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces

On a complete metric space X, we solve the problem of constructing an algorithm (in general, nonunique) of successive approximations from any point in space to a given closed subsetA. We give an estimate of the distance from an arbitrary initial point to the corresponding limit points. We consider three versions of the subset A: (1) A is the complete preimage of a closed subspace H under a mapping from X into the metric space Y; (2) A is the set of coincidence points of n (n > 1) mappings from X into Y; (3) A is the set of common fixed points of n mappings of X into itself (n = 1, 2, …). The problems under consideration are stated conveniently in terms of a multicascade, i.e., of a generalized discrete dynamical system with phase space X, translation semigroup equal to the additive semigroup of nonnegative integers, and the limit set A. In particular, in case (2) for n = 2, we obtain a generalization of Arutyunov’s theorem on the coincidences of two mappings. In case (3) for n = 1, we obtain a generalization of the contraction mapping principle.     

A Cameron-Storvick Theorem for Analytic Feynman Integrals on Product Abstract Wiener Space and Applications

In this paper we derive a Cameron-Storvick theorem for the analytic Feynman integral of functionals on product abstract Wiener space B ². We then apply our result to obtain an evaluation formula for the analytic Feynman integral of unbounded functionals on B ². We also present meaningful examples involving functionals which arise naturally in quantum mechanics.     

The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group

Let Mⁿ be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on Mⁿ of length at most 3 diameter(Mⁿ). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(Mⁿ), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of Mⁿ. If n=2, then this critical θ-graph is not only immersed but embedded.Mathematics Subject Classifications (2000). 53C23, 49Q10     

Polygonizations of point sets in the plane

We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons.We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n ⁴) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n ⁵) to compute a complete list ofO(n ⁴) nondegenerate star-shaped polygonizations of the set ofn points.     

A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation

We obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution.We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct. Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point.