Problème de Dirichlet pour une équation de Monge-Ampère réelle elliptique dégénérée en dimension

RÉSUMÉ. On considère dans un ouvert borné de , à bord régulier, le problème de Dirichlet

où , est positive et s'annule sur un ensemble fini de points de . On démontre alors sous certaines hypothèses sur et si est assez petit, que le problème (1) possède une solution convexe unique .

ABSTRACT. We consider in a bounded open set of , with regular boundary, the Dirichlet problem

where , is positive and vanishes on , a finite set of points in . We prove, under some hypothesis on and if is sufficiently small, that the problem (1) has a unique convex solution .

     

On the Diophantine equation : Higher-order recurrences

Let be a field of characteristic and let be a linear recurring sequence of degree in defined by the initial terms and by the difference equation

with . Finally, let be an element of . In this paper we are giving fairly general conditions depending only on on , and on under which the Diophantine equation

has only finitely many solutions . Moreover, we are giving an upper bound for the number of solutions, which depends only on . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

     

The Dirichlet problem and nondivergence harmonic measure

We consider the Dirichlet problem

for two second-order elliptic operators , , in a bounded Lipschitz domain . The coefficients belong to the space of bounded mean oscillation with a suitable small modulus. We assume that is regular in for some , , that is, for all continuous boundary data . Here is the surface measure on and is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients that will assure the perturbed operator to be regular in for some , .

     

On the capacity of sets of divergence associated with the spherical partial integral operator

In this article, we study the pointwise convergence of the spherical partial integral operator when it is applied to functions with a certain amount of smoothness. In particular, for , , , we prove that -quasieverywhere on , where is such that almost everywhere. A weaker version of this result in the range as well as some related localisation principles are also obtained. For and , we construct a function such that diverges everywhere.

     

The periodic Euler-Bernoulli equation

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

where the functions and are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by and . Here we develop a theory analogous to the theory of the Hill operator .

We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or -spectrum.

Our new analysis begins with a detailed study of the zeros of the function , for any given ``quasimomentum' , where is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to is , where is the discriminant and the period of ). We show that the multiplicity of any zero of can be one or two and (for some ) if and only if is also a zero of another entire function , independent of . Furthermore, we show that has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each -gap. If is a double zero of , it may happen that there is only one Floquet solution with quasimomentum ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.

Next we show that if is an open -gap of the pseudospectrum (i.e., ), then the Floquet matrix has a specific Jordan anomaly at and .

We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by the eigenvalues of this multipoint problem and show that is also characterized as the set of values of for which there is a proper Floquet solution such that .

We also show (Theorem 7) that each gap of the -spectrum contains exactly one and each -gap of the pseudospectrum contains exactly two 's, counting multiplicities. Here when we say ``gap' or ``-gap' we also include the endpoints (so that when two consecutive bands or -bands touch, the in-between collapsed gap, or -gap, is a point). We believe that can be used to formulate the associated inverse spectral problem.

As an application of Theorem 7, we show that if is a collapsed (``closed') -gap, then the Floquet matrix is diagonalizable.

Some of the above results were conjectured in our previous works. However, our conjecture that if all the -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.

     

Weighted estimates in for Laplace's equation on Lipschitz domains

Let , , be a bounded Lipschitz domain. For Laplace's equation in , we study the Dirichlet and Neumann problems with boundary data in the weighted space , where , is a fixed point on , and denotes the surface measure on . We prove that there exists such that the Dirichlet problem is uniquely solvable if , and the Neumann problem is uniquely solvable if . If is a domain, one may take . The regularity for the Dirichlet problem with data in the weighted Sobolev space is also considered. Finally we establish the weighted estimates with general weights for the Dirichlet and regularity problems.

     

Random variable dilation equation and multidimensional prescale functions

A random variable satisfying the random variable dilation equation , where is a discrete random variable independent of with values in a lattice and weights and is an expanding and -preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density which will satisfy a dilation equation

We have obtained necessary and sufficient conditions for the existence of the density and a simple sufficient condition for 's existence in terms of the weights Wavelets in can be generated in several ways. One is through a multiresolution analysis of generated by a compactly supported prescale function . The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when is a prescale function.

     

Non-solvability for a class of left-invariant second-order differential operators on the Heisenberg group

We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group , of the form

where is a complex matrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that and their commutator are linearly independent, we show that is not locally solvable, even in the presence of lower-order terms, provided that . In the case we show that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group a phenomenon first observed by Karadzhov and Müller in the case of It is interesting to notice that the analysis of the exceptional operators for the case turns out to be more elementary than in the case When the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.

     

A classification and examples of rank one chain domains

A chain order of a skew field is a subring of so that implies Such a ring has rank one if , the Jacobson radical of is its only nonzero completely prime ideal. We show that a rank one chain order of is either invariant, in which case corresponds to a real-valued valuation of or is nearly simple, in which case and are the only ideals of or is exceptional in which case contains a prime ideal that is not completely prime. We use the group of divisorial of with the subgroup of principal to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index of in Using the covering group of and the result that the group ring is embeddable into a skew field for a skew field, examples of rank one chain orders are constructed for each possible exceptional case.

     

Analytic models for commuting operator tuples on bounded symmetric domains

For a domain in and a Hilbert space of analytic functions on which satisfies certain conditions, we characterize the commuting -tuples of operators on a separable Hilbert space  such that is unitarily equivalent to the restriction of to an invariant subspace, where is the operator -tuple on the Hilbert space tensor product  . For the unit disc and the Hardy space , this reduces to a well-known theorem of Sz.-Nagy and Foias; for a reproducing kernel Hilbert space on such that the reciprocal of its reproducing kernel is a polynomial in and  , this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces for which ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) on a Cartan domain corresponding to the parameter in the continuous Wallach set, and reproducing kernel Hilbert spaces for which is a rational function. Further, we treat also the more general problem when the operator is replaced by ,  being a certain generalization of a unitary operator tuple. For the case of the spaces on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on , which seems to be of an independent interest.

     

One-dimensional asymptotic classes of finite structures

A collection of finite -structures is a 1-dimensional asymptotic class if for every and every formula , where :

(i)

There is a positive constant and a finite set such that for every and , either , or for some ,

(ii)

For every , there is an -formula , such that is precisely the set of with

One-dimensional asymptotic classes are introduced and studied here. These classes come equipped with a notion of dimension that is intended to provide for the study of classes of finite structures a concept that is central in the development of model theory for infinite structures. Connections with the model theory of infinite structures are also drawn.

     

Strongly indefinite functionals and multiple solutions of elliptic systems

We study existence and multiplicity of solutions of the elliptic system

where , is a smooth bounded domain and . We assume that the nonlinear term

where , , and . So some supercritical systems are included. Nontrivial solutions are obtained. When is even in , we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if 2$"> (resp. ). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.

     

Convolution roots of radial positive definite functions with compact support

A classical theorem of Boas, Kac, and Krein states that a characteristic function with for admits a representation of the form

where the convolution root is complex-valued with for . The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If is real-valued and even, can the convolution root be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of is obtained. Furthermore, the analogous problem for radially symmetric functions defined on is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turán's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if is a probability density on whose characteristic function vanishes outside the unit ball, then

where denotes the first positive zero of the Bessel function , and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in does not exist.

     

Hyponormality of trigonometric Toeplitz operators

In this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric Toeplitz operators, i.e., Toeplitz operators with trigonometric polynomial symbols . Our criterion involves the zeros of an analytic polynomial induced by the Fourier coefficients of . Moreover the rank of the selfcommutator of is computed from the number of zeros of in the open unit disk and in counting multiplicity.

     

The co-area formula for Sobolev mappings

We extend Federer's co-area formula to mappings belonging to the Sobolev class , , m$">, and more generally, to mappings with gradient in the Lorentz space . This is accomplished by showing that the graph of in is a Hausdorff -rectifiable set.

     

Bases for some reciprocity algebras I

For a complex vector space , let be the algebra of polynomial functions on . In this paper, we construct bases for the algebra of all highest weight vectors in , where and for all , and the algebra of highest weight vectors in .

     

On the essential commutant of QC

Let (QC) (resp. ) be the -algebra generated by the Toeplitz operators QC (resp. ) on the Hardy space of the unit circle. A well-known theorem of Davidson asserts that (QC) is the essential commutant of . We show that the essential commutant of (QC) is strictly larger than . Thus the image of in the Calkin algebra does not satisfy the double commutant relation. We also give a criterion for membership in the essential commutant of (QC).

     

Limits of interpolatory processes

Given distinct real numbers and a positive approximation of the identity , which converges weakly to the Dirac delta measure as goes to zero, we investigate the polynomials which solve the interpolation problem

with prescribed data . More specifically, we are interested in the behavior of when the data is of the form for some prescribed function . One of our results asserts that if is sufficiently nice and has sufficiently well-behaved moments, then converges to a limit which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is , where is an arbitrary finite subset of the integer lattice , as their degree goes to infinity.

     

The cyclic and simplicial cohomology of

Let be the unital semigroup algebra of . We show that the cyclic cohomology groups vanish when is odd and are one dimensional when is even (). Using Connes' exact sequence, these results are used to show that the simplicial cohomology groups vanish for . The results obtained are extended to unital algebras for some other semigroups of .