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 Epstein's Zeta-Function

Let , and let be Epstein's zeta-function of the form Q. It is proved that for |t| > C > 0 one has the estimate

The obstacle problem for Monge Ampere equation

We consider the following obstacle problem for Monge-Ampere equation and discuss the regularity of the free boundary . We prove that is if f is bounded away from 0 and , and it is C ^1,1 if .Received: 4 February 2003, Accepted: 3 March 2004, Published online: 16 July 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     

Singular minimizers of strictly polyconvex functionals in {\bf R}^{2 \times 2}

We give examples of non-C¹ one-homogeneous mappings and non-negative strictly polyconvex functions f such that where B denotes the unit ball in . Such u are therefore singular minimizers of the corresponding strictly polyconvex functionals in appropriate Sobolev spaces. Received: 12 January 2004, Accepted: 13 September 2004, Published online: 10 December 2004     

Infinitely many bound states for some nonlinear scalar field equations

In this paper we consider the problem in , where p > 2 and if N > 2. Assuming that the potential a(x) is a regular function such that 0$" align="middle" border="0"> and that verifies suitable decay assumptions, but not requiring any symmetry property on it, we prove that the problem has infinitely many solutions.Received: 3 December 2003, Accepted: 10 May 2004, Published online: 22 December 2004     

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     

A minimum problem with free boundary for a degenerate quasilinear operator

In this paper we prove regularity (near flat points) of the free boundary 0\}\cap\Omega$" align="middle" border="0"> in the Alt-Caffarelli type minimum problem for the p-Laplace operator: 0\}}\right)dx\rightarrow \min\qquad (1 Received: 3 June 2003, Accepted: 9 June 2004, Published online: 8 February 2005Mathematics Subject Classification (2000): 35R35, 35J60The first author is partially supported by NSF Grant DMS-0202801 and NSF CAREER Grant DMS-0239771     

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

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

$C^\infty$ 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

The Nehari manifold for a semilinear elliptic equation involving a sublinear term

The Nehari manifold for the equation for together with Dirichlet boundary conditions is investigated in the case where . Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as changes and show how this is linked to results on bifurcation from infinity which are associated with the problem.Received: 8 December 2003, Accepted: 10 May 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 35J20, 35J25     

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 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     

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

Isometries and direct decompositions of pseudo MV-algebras

In the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) there exists an internal direct decomposition of with commutative such that and for each x ∈ A. On the other hand, if is an internal direct decomposition of a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) with commutative, then the mapping g: A → A defined by is an isometry in and .      

Multiple Wick Product Chaos Processes

Let u(x) xR ^q be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p _x, (·) be the density of an R ^q valued canonical normal random variable with mean x and variance and let {G _x, ; (x, )R ^q ×[0,1 ]} be the mean zero Gaussian process with covariance

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     

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     

Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities

We consider a class of equations of the form By variational methods, we show the existence of families of positive solutions concentrating around local minima of the potential V(x), as . We do not require uniqueness of the ground state solutions of the associated autonomous problems nor the monotonicity of the function . We deal with asymptotically linear as well as superlinear nonlinearities.Received: 8 November 2003, Accepted: 18 November 2003, Published online: 2 April 2004Mathematics Subject Classification (2000): 35B25, 35J65, 58E05     

Stationary measures and rectifiability

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