A bijection between Littlewood-Richardson tableaux and rigged configurations

We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials labeled by a partition and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible -module of highest weight in the tensor product .     

On nonlocal calculation for inhomogeneous indefinite Neumann boundary value problems

This paper concerns with a family of inhomogeneous Neumann boundary value problems having indefinite nonlinearities which depend on a real parameter . We discuss the existence and the multiplicity of positive solutions with respect to . Developing the fibering method further, we can introduce a constructive concept of the calculation of certain nonlocal intervals , the so-called sufficient intervals of the existence. Then we are able to prove some new results on the existence and the multiplicity of positive solutions for .Received: 22 December 2003, Accepted: 29 January 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 35J70, 35J65, 47H17     

$C^\infty$ regularity of the free boundary for a two-dimensional optimal compliance problem

We study the regularizing effect of perimeter penalties for a problem of optimal compliance in two dimensions. In particular, we consider minimizers of

Boundary blow-up for a Brezis-Peletier problem on a singular domain

Let , , be a bounded domain as defined by Flucher, Garroni and Müller [6], which has a singular point such that the Robins function achieves its infimum at . Considering the elliptic problem in ; u = 0 on , with p = (N + 2)/(N-2), , and a minimizing solution of , concentrates at as goes to zero.Received: 15 September 2002, Accepted: 5 November 2002, Published online: 16 May 2003Mathematics Subject Classification: 35J65Angela Pistoia: The author is supported by M.U.R.S.T., project Metodi variazionali e topologici nello studio di fenomeni non lineari     

Inverse scattering up to smooth functions for the Dirac-ZS-AKNS system

We consider the Dirac-ZS-AKNS system (1) where (the space of functions with n derivatives in L ¹), (2) We consider for (1) the transition matrix and, in addition, for the case of the Dirac system (i.e. for the selfadjoint case the scattering matrix We can divide main results of the present work into three parts. I. We show that the inverse scattering transform and the inverse Fourier transform give the same solution, up to smooth functions, of the inverse scattering problem for (1). More preciseley, we show that, under condition (2) with , the following formulas are valid: (3) and, in addition, for the case of the Dirac system (4) where denotes the factor space. II. Using (3), (4), we give the characterization of the transition matrix and the scattering matrix for the case of the Dirac system under condition (2) with III. As applications of the results mentioned above, we show that 1) for any real-valued initial data , the Cauchy problem for the sh-Gordon equation has a unique solution such that and for any t > 0, 2) in addition, for , for such a solution the following formula is valid: where denotes the space of functions locally integrable with n derivatives. We give also a review of preceding results.     

On the relaxation and the Lavrentieff phenomenon for variational integrals with pointwise measurable gradient constraints

Relaxation problems for a functional of the type are analyzed, where is a bounded smooth open subset of and g is a Carathéodory function. The admissible functions u are forced to satisfy a pointwise gradient constraint of the type for a.e. being, for every , a bounded convex subset of , in general varying with x not necessarily in a smooth way. The relaxed functionals and of G obtained letting u vary respectively in , the set of the piecewise C ¹-functions in , and in in the definition of G are considered. For both of them integral representation results are proved, with an explicit representation formula for the density of . Examples are proposed showing that in general the two densities are different, and that the one of is not obtained from g simply by convexification arguments. Eventually, the results are discussed in the framework of Lavrentieff phenomenon, showing by means of an example that deep differences occur between and . Results in more general settings are also obtained.Received: 18 December 2002, Accepted: 18 November 2003, Published online: 16 July 2004Mathematics Subject Classification (2000): 49J45, 49J10, 49J53This work is part of the European Research Training Network Homogenization and Multiple Scales (HMS 2000), under contract HPRN-2000-00109. It is also part of the 2003-G.N.A.M.P.A. Project Metodi Variazionali per Strutture Sottili, Frontiere Oscillanti ed Energie Vincolate.     

When is the Fourier transform of an elementary function elementary?

Let F be a local field, a nontrivial unitary additive character of F, and V a finite dimensional vector space over F. Let us say that a complex function on V is elementary if it has the form , where , Q is a rational function (the phase function), are polynomials, and multiplicative characters of F. For generic , this function canonically extends to a distribution on V (if char(F) = 0). Occasionally, the Fourier transform of an elementary function is also an elementary function (the basic example is the Gaussian integral: k = 0, Q is a nondegenerate quadratic form). It is interesting to determine when exactly this happens. This question is the main subject of our study. In the first part of this paper we show that for or , if the Fourier transform of an elementary function with phase function -Q such that is another elementary function with phase function , then is the Legendre transform of Q (the "semiclassical condition"). We study properties and examples of phase functions satisfying this condition, and give a classification of phase functions such that both Q and are of the form f(x)/t, where f is a homogeneous cubic polynomial and t is an additional variable (this is one of the simplest possible situations). Unexpectedly, the proof uses Zak's classification theorem for Severi varieties.? In the second part of the paper we give a necessary and sufficient condition for an elementary function to have an elementary Fourier transform (in an appropriate "weak" sense) and explicit formulas for such Fourier transforms in the case when Q and are monomials, over any local field F. We also describe a generalization of these results to the case of monomials of norms of finite extensions of F. Finally, we generalize some of the above results (including Fourier integration formulas) to the case when and Q comes from a prehomogeneous vector space.     

Stationary measures and rectifiability

For integers , we consider -valued Radon measures on an open set which satisfy

Singular limits in Liouville-type equations

We consider the boundary value problem in a bounded, smooth domain in with homogeneous Dirichlet boundary conditions. Here 0,k(x) $$ " align="middle" border="0"> is a non-negative, not identically zero function. We find conditions under which there exists a solution which blows up at exactly m points as and satisfies . In particular, we find that if , 0 $" align="middle" border="0"> and is not simply connected then such a solution exists for any given Received: 11 February 2004, Accepted: 17 August 2004, Published online: 22 December 2004     

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

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.     

Maximum principles for submanifolds of arbitrary codimension and bounded mean curvature

We construct quadratic forms on which are subharmonic on any n-dimensional minimal submanifold in and, more generally, on submanifolds of bounded mean curvature. This leads to nonexistence results for connected n-dimensional minimal submanifolds in as well as to necessary conditions for the existence of connected submanifolds of bounded mean curvature with arbitrary codimension. Furthermore we discuss a barrier principle for n-dimensional submanifolds in of bounded mean curvature.Received: 11 November 2003, Accepted: 29 January 2004, Published online: 2 April 2004Mathematics Subject Classification (2000): 35J60, 49Q05, 53C42     

A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator

We prove a Sobolev inequality with remainder term for the imbedding , arbitrary, generalizing a corresponding result of Bianchi and Egnell for the case m = 1. We also show that the manifold of least energy solutions of the equation is a nondegenerate critical manifold for the corresponding variational integral. Finally we generalize the results of J. M. Coron on the existence of solutions of equations with critical exponent on domains with nontrivial topology to the biharmonic operator.Received: 21 March 2002, Accepted: 5 November 2002, Published online: 16 May 2003     

Approximating <Emphasis Type="Italic">J</Emphasis>-holomorphic curves by holomorphic ones

Given an almost complex structure J in a cylinder of (p > 1) together with a compatible symplectic form and given an arbitrary J-holomorphic curve without boundary in that cylinder, we construct an holomorphic perturbation of , for the canonical complex structure J ₀ of , such that the distance between these two curves in W ^1,2 and norms, in a sub-cylinder, are controled by quantities depending on J, and by the area of only. These estimates depend neither on the topology nor on the conformal class of . They are key tools in the recent proof of the regularity of 1-1 integral currents in [RT].Received: 2 October 2003, Accepted: 18 November 2003, Published online: 25 February 2004     

Some improved Caffarelli-Kohn-Nirenberg inequalities

For 1 < p < N and we obtain the following improved Hardy-Sobolev Inequalities where 1 < q < p and if , if , for some positive constant . Also we give an alternative proof of the optimal improved inequality for p = 2 by Wang-Willem in [16]. Received: 2 February 2004, Accepted: 12 July 2004, Published online: 3 September 2004 Mathematics Subject Classification (2000): 35J20, 35P05, 35R05, 46E30, 46E35 Partially supported by Project BFM2001-0183     

Dirichlet and Bergman Spaces of Holomorphic Functions on the Unit Ball of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let B denote the unit ball in ⁿ, n 1, and let and denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces , , and weighted Bergman spaces , , , of holomorphic functions f on B for which and respectively are finite, where and The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and .(a) If for some , then for all p, , with .(b) If for some p, , then for all with . Combining Theorem 1 with previous results of the author we also obtain the following.Theorem 2. Suppose is holomorphic in B. If for some p, , and , then . Conversely, if for some p, , then the series in * converges.     

A new approach to Sobolev spaces and connections to $\mathbf\Gamma$-convergence

This is a follow-up of a paper of Bourgain, Brezis and Mironescu [2]. We study how the existence of the limit

Inequalities for irregular Gabor frames

In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if is a Gabor frame for with frame bounds A and B, then the following two inequalities hold: and . In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form , where Δ _k and Λ _k are arbitrary sequences of points in and , 1 ≤ k ≤ r. Corresponding author for second author Authors’ address: Lili Zang and Wenchang Sun, Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China     

Minkowski Arrangements of Spheres

Let be a non-negative number not greater than 1. Consider an arrangement of (not necessarily congruent) spheres with positive homogenity in the n-dimensional Euclidean space, i.e., in which the infimum of the radii of the spheres divided by the supremum of the radii of the spheres is a positive number. With each sphere S of associate a concentric sphere of radius times the radius of S. We call this sphere the -kernel of S. The arrangement is said to be a Minkowski arrangement of order if no sphere of overlaps the -kernel of another sphere. The problem is to find the greatest possible density of n-dimensional Minkowski sphere arrangements of order . In this paper we give upper bounds on for .     

A new example of singular height transition for capillary surfaces

Finn and Kosmodemyanskii, Jr. gave an example of a domain containing a disk , and of a family of domains converging to as , such that the heights u ^t of capillary surfaces in vertical tubes with the sections in a gravity field g satisfy for every , but for which u ¹< u ⁰ over for all g > 0. In subsequent work, Finn and Lee characterized the most general convex that leads to such a discontinuous transition when is a disk. It has been suggested that the cause for this curious behavior is related to the fact that in all cases considered the boundaries of the have a discontinuity in their curvatures, that is bounded below in magnitude. In the present note we present an alternative form of the example, in which the domains are disks concentric to . Thus, the limited smoothness in the original example of the convergence to of the approxim ating domains cannot be viewed as the root cause of the anomaly. The procedure presented here leads to explicit bounds, which were not available in the earlier forms of the example.Received: 3 September 2002, Accepted: 17 February 2003, Published online: 1 July 2003Mathematics Subject Classification: 76B45, 53A10, 49Q10