Compensated compactness for nonlinear homogenization and reduction of dimension

We study the limit behaviour of some nonlinear monotone equations, such as: , in a domain which is thin in some directions (e.g. is a plate or a thin cylinder). After rescaling to a fixed domain , the above equation is transformed into: , with convenient operators and . Assuming that and the inverse of have particular forms and satisfy suitable compensated compactness assumptions, we prove a closure result, that is we prove that the limit problem has the same form. This applies in particular to the limit behaviour of nonlinear monotone equations in laminated plates.Received: 16 October 2002, Accepted: 12 June 2003, Published online: 22 September 2003Mathematics Subject Classification (2000): 35B27, 35B40, 74Q15     

Comparison of potential theoretic properties of rough domains

We discuss the relationships between the notions of intrinsic ultracontractivity, the parabolic Harnack principle, compactness of the 1-resolvent of the Neumann Laplacian, and the non-trap property for Euclidean domains with finite Lebesgue measure. In particular, we give an answer to an open problem raised by Davies and Simon in 1984 about the possible relationship between intrinsic ultracontractivity for the Dirichlet Laplacian in a domain and compactness of the 1-resolvent of the Neumann Laplacian in .

     

The Berezin transform and radial operators

We analyze the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a special class of operators that we call radial operators, an oscilation criterion is a sufficient condition under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit circle. We further study a special class of radial operators, i.e., Toeplitz operators with a radial symbol.

     

Non radial positive solutions for the Hénon equation with critical growth

We study the Dirichlet problem in a ball for the Hénon equation with critical growth and we establish, under some conditions, the existence of a positive, non radial solution. The solution is obtained as a minimizer of the quotient functional associated to the problem restricted to appropriate subspaces of H₀¹ invariant for the action of a subgroup of . Analysis of compactness properties of minimizing sequences and careful level estimates are the main ingredients of the proof. Received: 18 October 2003, Accepted: 5 July 2004, Published online: 3 September 2004 Mathematics Subject Classification (2000): 35J20, 35B33 This research was supported by MIUR, Project "Variational Methods and Nonlinear Differential Equations".     

A global compactness result for singular elliptic problems involving critical Sobolev exponent

Let be a bounded domain such that . Let be a (P.S.) sequence of the functional . We study the limit behaviour of and obtain a global compactness result.

     

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     

Minimal nodal solutions of the pure critical exponent problem on a symmetric domain

We establish existence of nodal solutions to the pure critical exponent problem in u = 0 on where a bounded smooth domain which is invariant under an orthogonal involution of We extend previous results for positive solutions due to Coron, Dancer, Ding, and Passaseo to existence and multiplicity results for solutions which change sign exactly once.Received: 4 April 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000): 35J65, 35J20Research partially supported by PAPIIT, UNAM, under grant IN110902-3.     

An approach to the spectrum structure of Dirac operators by the local-compactness method

The purpose of this paper is to investigate the spectra of the Dirac operator . The local compactness of is shown under some assumption on . This method enables us to prove that if as , then and to give a significant sufficient condition that or has a purely discrete spectrum.

     

On regular polytope numbers

Lagrange proved a theorem which states that every nonnegative integer can be written as a sum of four squares. This result can be generalized as the polygonal number theorem and the Hilbert-Waring problem. In this paper, we shall generalize Lagrange's sum of four squares theorem further. To each regular polytope in a Euclidean space, we will associate a sequence of nonnegative integers which we shall call regular polytope numbers, and consider the problem of finding the order of the set of regular polytope numbers associated to .

     

Randomised circular means of Fourier transforms of measures

We explore decay estimates for circular means of the Fourier transform of a measure on in terms of its -dimensional energy. We find new upper bounds for the decay exponent. We also prove sharp estimates for a certain family of randomised versions of this problem.

     

About the sharpness of the stability estimates in the Kreiss matrix theorem

One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices under consideration. This so-called resolvent condition is known to imply, for all , the upper bounds and . Here is the spectral norm, is the constant occurring in the resolvent condition, and the order of is equal to .

It is a long-standing problem whether these upper bounds can be sharpened, for all fixed 1$">, to bounds in which the right-hand members grow much slower than linearly with and with , respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each 0$">, there are fixed values 0, K>1$"> and a sequence of matrices , satisfying the resolvent condition, such that for .

The result proved in this paper is also relevant to matrices whose -pseudospectra lie at a distance not exceeding from the unit disk for all 0$">.

     

A tall space with a small bottom

We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if , then there is such a space of height with only many isolated points. This implies that there is a locally compact scattered space of height with isolated points in ZFC, solving an old problem of the first author.

     

On a proposed characterization of Schatten-class composition operators

For an analytic function which maps the open unit disc to itself, let be the operator of composition with on the Bergman space . It has been a longstanding problem to determine whether or not the membership of in the Schatten class , , is equivalent to the condition that the function has a finite integral with respect to the Möbius-invariant measure on . We show that the answer is negative when .

     

Infinitely many radial solutions of a variational problem related to dispersion-managed optical fibers

We consider a non-local variational problem whose critical points are related to bound states in certain optical fibers. The functional is given by , and relying on the regularizing properties of the solution to the free Schrödinger equation, it will be shown that has infinitely many critical points.

     

A computational approach for solving

We present a computational approach for finding all integral solutions of the equation for even values of . By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for assuming the Generalized Riemann Hypothesis, and for unconditionally.

     

Full-wave analysis of dielectric waveguides at a given frequency

New variational formulation to compute propagation constants is proposed. Based on it, vector finite element method is proved to exclude spurious modes provided finite elements possess discrete compactness property. Convergence analysis is conducted in the framework of collectively compact operators. Reported theoretical results apply to a wide class of vector finite elements including two families of Nedelec and their generalization, the -edge elements. Numerical experiments fully support theoretical estimates for convergence rates.

     

A free boundary problem for a singular system of differential equations: An application to a model of tumor growth

In this paper we consider a free boundary problem for a nonlinear system of two ordinary differential equations, one of which is singular at some points, including the initial point . Because of the singularity at , the initial value problem has a one-parameter family of solutions. We prove that there exists a unique solution to the free boundary problem. The proof of existence employs two ``shooting' parameters. Analysis of the profiles of solutions of the initial value problem and tools such as comparison theorems and weak limits of solutions play an important role in the proof. The system considered here is motivated by a model in tumor growth, but the methods developed should be applicable to more general systems.

     

Newman polynomials with prescribed vanishing and integer sets with distinct subset sums

We study the problem of determining the minimal degree of a polynomial that has all coefficients in and a zero of multiplicity at . We show that a greedy solution is optimal precisely when , mirroring a result of Boyd on polynomials with coefficients. We then examine polynomials of the form , where is a set of positive odd integers with distinct subset sums, and we develop algorithms to determine the minimal degree of such a polynomial. We determine that satisfies inequalities of the form . Last, we consider the related problem of finding a set of positive integers with distinct subset sums and minimal largest element and show that the Conway-Guy sequence yields the optimal solution for , extending some computations of Lunnon.