Multiple solutions of constant sign for nonlinear nonsmooth eigenvalue problems near resonance

In this paper we study a class of nonlinear elliptic eigenvalue problems driven by the p-Laplacian and having a nonsmooth locally Lipschitz potential. We show that as the parameter approaches (= the principal eigenvalue of ) from the right, the problem has three nontrivial solutions of constant sign. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions. In the process of the proof we also establish a generalization of a recent result of Brezis and Nirenberg for C₀¹ versus W₀^1,p minimizers of a locally Lipschitz functional. In addition we prove a result of independent interest on the existence of an additional critical point in the presence of a local minimizer of constant sign. Finally by restricting further the asymptotic behavior of the potential at infinity, we show that for all the problem has two solutions one strictly positive and the other strictly negative.Received: 7 January 2003, Accepted: 12 May 2003, Published online: 4 September 2003Mathematics Subject Classification (2000): 35J20, 35J85, 35R70     

Numerical index and renorming

We study the numerical index of a Banach space from the isomorphic point of view, that is, we investigate the values of the numerical index which can be obtained by renorming the space. The set of these values is always an interval which contains in the real case and in the complex case. Moreover, for ``most' Banach spaces the least upper bound of this interval is as large as possible, namely .

     

Diophantine approximation by algebraic hypersurfaces and varieties

Questions on rational approximations to a real number can be generalized in two directions. On the one hand, we may ask about ``approximation' to a point in by hyperplanes defined over the rationals. That is, we seek hyperplanes with small distance from the given point. On the other hand, following Wirsing, we may ask about approximation to a real number by real algebraic numbers of degree at most .

The present paper deals with a common generalization of both directions, namely with approximation to a point in by algebraic hypersurfaces, or more generally algebraic varieties defined over the rationals.

     

A Relaxed Approximate Proximal Point Algorithm

For a maximal monotone operator T, a well-known overrelaxed point algorithm is often used to find the zeros of T. In this paper, we enhance the algorithm to find a point in , where is a given closed convex set. In the inexact case of our modified relaxed proximal point algorithm, we give a new criterion. The convergence analysis is quite easy to follow.     

Symmetric functions in noncommuting variables

Consider the algebra of formal power series in countably many noncommuting variables over the rationals. The subalgebra of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as well as investigating their properties.

     

The loss of tightness of time distributions for homeomorphisms of the circle

For a minimal circle homeomorphism we study convergence in law of rescaled hitting time point process of an interval of length 0$">. Although the point process in the natural time scale never converges in law, we study all possible limits under a subsequence. The new feature is the fact that, for rotation numbers of unbounded type, there is a sequence going to zero exhibiting coexistence of two non-trivial asymptotic limit point processes depending on the choice of time scales used when rescaling the point process. The phenomenon of loss of tightness of the first hitting time distribution is an indication of this coexistence behaviour. Moreover, tightness occurs if and only if the rotation number is of bounded type. Therefore tightness of time distributions is an intrinsic property of badly approximable irrational rotation numbers.

     

Non-additivity for triple point numbers on the connected sum of surface-knots

Any surface-knot in 4-space can be projected into 3-space with a finite number of triple points, and its triple point number, , is defined similarly to the crossing number of a classical knot. By definition, we have for the connected sum. In this paper, we give infinitely many pairs of surface-knots for which this equality does not hold.

     

Affine pseudo-planes and cancellation problem

We define affine pseudo-planes as one class of -homology planes. It is shown that there exists an infinite-dimensional family of non-isomorphic affine pseudo-planes which become isomorphic to each other by taking products with the affine line . Moreover, we show that there exists an infinite-dimensional family of the universal coverings of affine pseudo-planes with a cyclic group acting as the Galois group, which have the equivariant non-cancellation property. Our family contains the surfaces without the cancellation property, due to Danielewski-Fieseler and tom Dieck.

     

Revêtements et isométries pour la métrique infinitésimale de Kobayashi

In this paper, we prove that, under some hypothesis on the domains, if a holomorphic mapping is an isometry for the Kobayashi infinitesimal metric at a point, it is a covering map. In the case , we prove, in certain cases, that is an analytic isomorphism.

     

Alternate signs Banach-Saks property and real interpolation of operators

In the space of bounded linear operators acting between Banach spaces we define a seminorm vanishing on the subspace of operators having the alternate signs Banach-Saks property. We obtain logarithmically convex-type estimates of the seminorm for operators interpolated by the Lions-Peetre real method. In particular, the estimates show that the alternate signs Banach-Saks property is inherited from a space of an interpolation pair to the real interpolation spaces with respect to for all and .

     

Spherical harmonic projectors

The harmonic projection (HP), which is implicit in the numerous harmonic transforms between physical and spectral spaces, is responsible for the reliability of the spectral method for modeling geophysical phenomena. As currently configured, the HP consists of a forward transform from physical to spectral space (harmonic analysis) immediately followed by a harmonic synthesis back to physical space. Unlike its Fourier counterpart in Cartesian coordinates, the HP does not identically reconstruct the original function on the surface of the sphere but rather replaces it with a weighted least-squares approximation. The importance of the HP is that it uniformly resolves waves on the surface of the sphere and therefore eliminates high frequencies that are artificially induced by the clustering of grid points in the neighborhood of the poles. The HP also maintains spectral accuracy when combined with the double Fourier method. Originally the HP required storage where is the number of latitudinal points. However, this was recently reduced to using an algorithm that also provided a savings of up to 50 percent in compute time. The HP was also generalized to an arbitrary latitudinal distribution of points. However, the HP as a composite of analysis and synthesis can be subject to considerable error depending on the point distribution. Here we define a variant of the traditional HP that is well conditioned, with condition number 1, for any point distribution. In addition, storage requirements are further reduced because the projections corresponding to all longitudinal wave numbers are expressed in terms of a single orthogonal matrix.

     

Modular equations for hyperelliptic curves

We define modular equations describing the -torsion subgroups of the Jacobian of a hyperelliptic curve. Over a finite base field, we prove factorization properties that extend the well-known results used in Atkin's improvement of Schoof's genus 1 point counting algorithm.

     

Group actions on one-manifolds, II: Extensions of Hölder's Theorem

This self-contained paper is part of a series seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in as those groups whose elements have at most one fixed point.

     

The index of a critical point for densely defined operators of type in Banach spaces

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type acting from a real, reflexive and separable Banach space into This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in 2,$"> illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type Applications to nonlinear Dirichlet problems have appeared elsewhere.

     

The double-base number system and its application to elliptic curve cryptography

We describe an algorithm for point multiplication on generic elliptic curves, based on a representation of the scalar as a sum of mixed powers of and . The sparseness of this so-called double-base number system, combined with some efficient point tripling formulae, lead to efficient point multiplication algorithms for curves defined over both prime and binary fields. Side-channel resistance is provided thanks to side-channel atomicity.

     

On the addition of convex sets in the hyperbolic plane

Analogue to the definition $K + L := \bigcup_{x\in K}(x + L)$ of the Minkowski addition in the euclidean geometry it is proposed to define the (noncommutative) addition $K \vdash L := \bigcup_{0\, \leqsl\, \rho\,\leqsl\, a(\varphi),0\,\leqsl\,\varphi\,<\, 2\pi}T_{\rho}^{(\varphi)}(L)$ for compact, convex and smoothly bounded sets K and L in the hyperbolic plane $\Omega$ (Kleins model). Here $\rho = a(\varphi)$ is the representation of the boundary $\partial$ K in geodesic polar coordinates and $T_{\rho}^{(\varphi)}$ is the hyperbolic translation of $\Omega$ of length $\rho$ along the line through the origin o of direction $\varphi$. In general this addition does not preserve convexity but nevertheless we may prove as main results: (1) $o \in$ int $K, o \in$ int L and K,L horocyclic convex imply the strict convexity of $K \vdash L$, and (2) in this case there exists a hyperbolic mixed volume $V_h(K,L)$ of K and L which has a representation by a suitable integral over the unit circle.     

Self-similar sets with an open set condition and great variety of overlaps

For a very simple family of self-similar sets with two pieces, we prove, using a technique of Solomyak, that the intersection of the pieces can be a Cantor set with any dimension in as well as a finite set of any cardinality . The main point is that the open set condition is fulfilled for all these examples.

     

Fixed point results for generalized contractions in gauge spaces and applications

In this paper, we present fixed point results for generalized contractions defined on a complete gauge space . Also, we consider families of generalized contractions where is closed and can have empty interior. We give conditions under which the existence of a fixed point for some imply the existence of a fixed point for every . Finally, we apply those results to infinite systems of first order nonlinear differential equations and to integral equations on the real line.

     

<Emphasis Type="Italic">L</Emphasis>-convex-concave body in ${\boldmath{$\mathbb{R}P^3$}}$ contains a line

We define a class of L-convex-concave subsets of ${\boldmath{$\mathbb{R}P^3$}}$, where L is a projective line in ${\boldmath{$\mathbb{R}P^3$}}$. These are sets whose sections by any plane containing L are convex and concavely depend on this plane. We prove a version of Arnolds conjecture for these sets, namely we prove that each such set contains a line.     

Critical points of the area functional of a complex closed curve on the manifold of Kähler metrics

We consider a compact complex manifold of dimension that admits Kähler metrics and we assume that is a closed complex curve. We denote by the space of classes of Kähler forms that define Kähler metrics of volume 1 on and define by . We show how the Riemann-Hodge bilinear relations imply that any critical point of is the strict global minimum and we give conditions under which there is such a critical point : A positive multiple of is the Poincaré dual of the homology class of . Applying this to the Abel-Jacobi map of a curve into its Jacobian, , we obtain that the Theta metric minimizes the area of within all Kähler metrics of volume 1 on .