A sharp attainment result for nonconvex variational problems

We consider the problem of minimizing autonomous, multiple integrals like

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     

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

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     

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.

     

A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold

A class of minimal almost complex submanifolds of a Riemannian manifold with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kähler submanifold is defined. Any Kähler submanifold is pluriminimal. In the case of a quaternionic Kähler manifold of non zero scalar curvature, in particular, when is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of Kähler submanifolds M²ⁿ of maximal possible dimension 2n. More precisely, we prove that any such Kähler submanifold M²ⁿ of is the projection of a holomorphic Legendrian submanifold of the twistor space of , considered as a complex contact manifold with the natural holomorphic contact structure . Any Legendrian submanifold of the twistor space is defined by a generating holomorphic function. This is a natural generalization of Bryants construction of superminimal surfaces in S⁴=P¹. Mathematics Subject Classification (1991) Primary: 53C40; Secondary: 53C55     

Partial regularity of minimizers of polyconvex variational integrals

We prove partial regularity of vector-valued minimizers u of the polyconvex variational integral , where stands for the minors of the gradient Du. For the integrand, we assume f to be a continuous function of class C ², strictly convex and of polynomial growth in the minors, and g to be a bounded Carathéodory function. We do not employ a Caccioppoli inequality.Received: 19 March 2002, Accepted: 24 October 2002, Published online: 16 May 2003Mathematics Subject Classification (2000): 49N60, 35J50     

Higher-order energy expansions and spike locations

We consider the following singularly perturbed semilinear elliptic problem: where is a bounded domain in R ^N with smooth boundary , is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional defined by where . Ni and Takagi ([29, 30]) proved that for a single boundary spike solution , the following asymptotic expansion holds: where c ₁ > 0 is a generic constant, is the unique local maximum point of and is the boundary mean curvature function at . In this paper, we obtain a higher-order expansion of where c ₂, c ₃ are generic constants and is the scalar curvature at . In particular c ₃ > 0. Some applications of this expansion are given.Received: 14 January 2003, Accepted: 28 July 2003, Published online: 15 October 2003Mathematics Subject Classification (2000): Primary 35B40, 35B45; Secondary 35J25     

Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations

Let be a bounded starshaped domain. In this note we consider critical points of the functional

where of class satisfies the natural growth

for some and 0$">, is suitably rank-one convex and in addition is strictly quasiconvex at . We establish uniqueness results under the extra assumption that is stationary at with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Sverák (2003).

     

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.     

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     

Dirichlet problem with indefinite nonlinearities

We consider the following nonlinear elliptic equation in a bounded domain with the Dirichlet boundary condition, and , g₁(u)u and g₂(u)u are positive for |u| > > 1. Some existence results are given for superlinear g₁ and g₂ via the Morse theory.Received: 16 Januray 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000): 35J20, 35J25, 58E05Parts of the work were completed while the authors were visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. The authors thank the hospitality of ICTP. Both authors are supported by NSFC, RFDP, MCME, the second author is also supported by the Foundation for University Key Teacher of the Ministry of Education of China and the 973 project of the Ministry of Science and Technology of China.     

Application of a Riesz-type formula to weighted Bergman spaces

Let denote the unit disk in the complex plane. We consider a class of superbiharmonic weight functions whose growth are subject to the condition for some constant . We first establish a Reisz-type representation formula for , and then use this formula to prove that the polynomials are dense in the weighted Bergman space with weight .

     

Endpoint estimates for the circular maximal function

We consider the problem of endpoint estimates for the circular maximal function defined by

where is the normalized surface area measure on . Let be the closed triangle with vertices . We prove that for , there is a constant such that Furthermore, .

     

Some prime factorization results for type II<Subscript>1</Subscript> factors

We prove several unique prime factorization results for tensor products of type II₁ factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1. In particular, we show that if is isomorphic to a subfactor in , for some 2r_i,s_j, then mn. Mathematics Subject Classification (2000) Primary 46L10; Secondary 20F67     

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     

Tight frame oversampling and its equivalence to shift-invariance of affine frame operators

Let generate a tight affine frame with dilation factor , where , and sampling constant (for the zeroth scale level). Then for , oversampling (or oversampling by ) means replacing the sampling constant by . The Second Oversampling Theorem asserts that oversampling of the given tight affine frame generated by preserves a tight affine frame, provided that is relatively prime to (i.e., ). In this paper, we discuss the preservation of tightness in oversampling, where (i.e., and ). We also show that tight affine frame preservation in oversampling is equivalent to the property of shift-invariance with respect to of the affine frame operator defined on the zeroth scale level.

     

Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity

We study the conformal scalar curvature problem

where is a continuous function. We show that a necessary and sufficient condition on for this problem to have positive solutions which are arbitrarily large at is that be less than 1 on a sequence of points in which tends to .

     

Primal separation algorithms

Given an integer polyhedron , an integer point , and a point , the primal separation problem is the problem of finding a linear inequality which is valid for P _I , violated by x ^*, and satisfied at equality by . The primal separation problem plays a key role in the primal approach to integer programming. In this paper we examine the complexity of primal separation for several well-known classes of inequalities for various important combinatorial optimization problems, including the knapsack, stable set and travelling salesman problems.Received: November 2002, Revised: March 2003,     

Infinitesimal study of a factorization of the Prym map

The Prym map factors through a space ={X} of intrinsically polarized varieties, namely , where is a connected étale double cover of a smooth curve C of genus g, consists of the set of even precanonical effective divisors on , and (P,) is the principally polarized Prym variety associated to . X is a connected, reduced local complete intersection of (pure) dimension g-1, and when C is non hyperelliptic X is normal and irreducible. By analogy with the proofs of the classical Torelli theorem for curves by Andreotti and by Andreotti-Mayer and Green, which factor the Jacobi map through a symmetric product of the curve, the present factorization may be used to attack the Torelli problem for Prym varieties. In [19] we have shown that X determines the Prym variety (P,), as the Albanese variety of X, and that X also determines the double cover , at least when C is non hyperelliptic and the codimension of sing in P is at least 5. The next challenge in this approach to the Torelli problem is to analyze the infinitesimal structure of these maps.The goal of the present paper is to show the first map is unramified when C is non hyperelliptic, i.e. that a first order deformation of which induces the trivial first order deformation of X, is already trivial on . (This question was studied for g=3 by H. Yin [23].) We do this as follows for g3. There is a map from to a curve of effective Cartier divisors on X. We prove that if C is non hyperelliptic, this map is an isomorphism from onto a smooth connected component of the Hilbert scheme of X. This is an analogue of Prop. 4.1. b), p.334, in [7], (that the set , is a connected component of Hilb(C^(g-1)) isomorphic to C).Then we deduce that if a first order deformation of induces the trivial deformation of X, the deformation of is isomorphic to the trivial deformation of the curve in Hilb(X). It follows that the original deformation of is trivial. The complementary question of whether every first order deformation of X comes from a first order deformation of , analogous to Thm. 3.6 of [7], is proved in [23] for g=3 and C non hyperelliptic, but remains open for g4 at the time of writing. We will work throughout over the complex numbers, and will generally assume the base curve C is smooth and non hyperelliptic, although some results are true more generally. Dedicated in memory of Fabio BardelliMathematics Subject Classification (2000) 14H40, 14K