The monopolist’s problem: existence, relaxation, and approximation

We study a variational problem arising from a generalization of an economic model introduced by Rochet and Choné in [5]. In this model a monopolist proposes a set Y of products with price list . Each rational consumer chooses which product to buy by solving a personal minimum problem, taking into account his/her tastes and economic possibilities. The monopolist looks for the optimal price list which minimizes costs, hence maximizes the profit. This leads to a minimum problem for functionals (the “pessimistic cost expectation”) and (the “optimistic cost expectation”), which are in turn defined through two nested variational problems. We prove that the minimum of exists and coincides with the infimum of . We also provide a variational approximation of by smooth functionals defined in finite dimensional Euclidean spaces.Received: 2 March 2004, Accepted: 19 October 2004, Published online: 22 December 2004Mathematics Subject Classification (2000): 49J45, 91B     

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     

A note about the generalized Hardy-Sobolev inequality with potential in L^{{p,d}} {\left( {\mathbb{R}^{n} } \right)}

We present a generalized version of the Hardy-Sobolev inequality, in which the homogeneous potential is replaced by any potential V belonging to the Lorentz space . We show that the best constant in these inequalities is achieved provided that where . We also analyze the limit case . Finally an application to a non-linear eigenvalues problem with rough potentials is presented.Received: 19 September 2004, Accepted: 15 November 2004, Published online: 22 December 2004     

On the \Gamma-limit of the Mumford-Shah functional

We study by means of -convergence the asymptotics of the rescaled Mumford-Shah functional

On the minimizers of the relaxed energy functional of mappings from higher dimensional balls into S 2

We prove a new approximation theorem, which enables us to show that the relaxed energy of Sobolev mappings u from higher dimensional balls into S² is given by , provided their singular set is of Lebesgue measure zero. Here is the mass of the minimal integer multiplicity connection associated to the singularity current S_u of u. Using this approximation theorem, we prove a partial regularity theorem for minimizers of the relaxed energy functional.Received: 5 May 2004, Accepted: 19 October 2004, Published online: 10 December 2004     

A Helly-type theorem for countable intersections of starshaped sets

Let k and d be fixed integers, 0kd, and let be a collection of sets in If every countable subfamily of has a starshaped intersection, then is (nonempty and) starshaped as well. Moreover, if every countable subfamily of has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004     

Optimal lifting for BV(S 1,S 1)

For each , we solve the following variational problem and we show that .Received: 4 November 2003, Accepted: 10 May 2004, Published online: 16 July 2004Radu Ignat: Radu.Ignat@ens.fr     

The Dirichlet energy of mappings from ${\bf B^3}$ into a manifold: density results and gap phenomenon

Weak limits of graphs of smooth maps with equibounded Dirichlet integral give rise to elements of the space . We assume that the 2-homology group of has no torsion and that the Hurewicz homomorphism is injective. Then, in dimension n = 3, we prove that every element T in , which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs {u _k } with Dirichlet energies converging to the energy of T. We also show that the injectivity hypothesis on the Hurewicz map cannot be removed. We finally show that a similar topological obstruction on the target manifold holds for the approximation problem of the area functional.Received: 9 May 2003, Accepted: 5 June 2003, Published online: 25 February 2004     

Integral Equations on Function Spaces and Dichotomy on the Real Line

The purpose of this paper is to give new and general characterizations for uniform dichotomy and uniform exponential dichotomy of evolution families on the real line. We consider two general classes denoted and and we prove that if V,W are Banach function spaces with and , then the admissibility of the pair for an evolution family implies the uniform dichotomy of . In addition, we consider a subclass and we prove that if , then the admissibility of the pair implies the uniform exponential dichotomy of the family . This condition becomes necessary if . Finally, we present some applications of the main results.     

Indecomposable Elements in K1 of a Smooth Projective Variety

Let X be a smooth projective variety over the complex numbers. We consider the cohomology of the sheaves and arising from Deligne–Beilinson cohomology and the Hodge filtration on the singular cohomology of X. We show that one can identify with the image of the truncated regulator map c¯2,1. In particular, this implies that is countable. Since this group is a direct summand of coker , this gives a partial answer to Voisin's conjecture that cocker() is countable. In the case of X a surface, we prove that the Albanese kernel T(X) is isomorphic to the group of global sections of if and only if pg=0.     

Regularity and relaxed problems of minimizing biharmonic maps into spheres

For and , we show that any minimizing biharmonic map from to S^k is smooth off a closed set whose Hausdorff dimension is at most n-5. When n = 5 and k = 4, for a parameter we introduce a -relaxed energy of the Hessian energy for maps in so that each minimizer of is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of for .Received: 5 April 2004, Accepted: 19 October 2004, Published online: 10 December 2004     

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     

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.     

$C^\infty$ 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

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.     

Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

Symmetry of positive solutions of an almost-critical problem in an annulus

We consider the subcritical problem & 0 & \qquad\textrm{in} \; A\\ u & = & 0 & \qquad\textrm{on} \; \delta A\\ \end{array} \right.$$" align="middle" border="0"> where A is an annulus in , , is the critical Sobolev exponent and 0$" align="middle" border="0"> is a small parameter. We prove that solutions of (I) which concentrate at one or two points are axially symmetric.Received: 7 July 2003, Accepted: 10 May 2004, Published online: 16 July 2004Filomena Pacella: Research supported by MIUR, project Variational Methods and Nonlinear Differential Equations.     

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     

Generalized de Bruijn Cycles

For a set of integers , we define a q-ary -cycle to be an assignment of the symbols 1 through q to the integers modulo q ⁿ so that every word appears on some translate of . This definition generalizes that of de Bruijn cycles, and opens up a multitude of questions. We address the existence of such cycles, discuss reduced cycles (ones in which the all-zeroes string need not appear), and provide general bounds on the shortest sequence which contains all words on some translate of . We also prove a variant on recent results concerning decompositions of complete graphs into cycles and employ it to resolve the case of completely.AMS Subject Classification: 94A55, 05C70.