The Jacobian and the Ginzburg-Landau energy

We study the Ginzburg-Landau functional for , where U is a bounded, open subset of . We show that if a sequence of functions satisfies , then their Jacobians are precompact in the dual of for every . Moreover, any limiting measure is a sum of point masses. We also characterize the -limit of the functionals , in terms of the function space B2V introduced by the authors in [16,17]: we show that I(u) is finite if and only if , and for is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if then the Jacobians are again precompact in for all , and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation of the Jacobian measure is a lower bound for the limit of the Ginzburg-Landau functional. Received: 15 December 2000 / Accepted: 23 January 2001 / Published online: 25 June 2001     

Numerical exterior algebra and the compound matrix method

Summary. The compound matrix method, which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k=2,3 dimensional subspaces of , by using compound matrices as coordinates, is reformulated in a coordinate-free way using exterior algebra spaces, . This formulation leads to a general framework for studying systems of differential equations on k-dimensional subspaces. The framework requires the development of several new ideas: the role of Hodge duality and the Hodge star operator in the construction, an efficient strategy for constructing the induced differential equations on , general formulation of induced boundary conditions, the role of geometric integrators for preserving the manifold of k-dimensional subspaces – the Grassmann manifold, , and a formulation for induced systems on an unbounded interval. The numerical exterior algebra framework is most advantageous for numerical solution of differential eigenvalue problems on unbounded domains, where there are significant difficulties in setting up matrix discretizations. The formulation is presented for k-dimensional subspaces of systems on with k and n arbitrary, and examples are given for the cases of k=2 and n=4, and k=3 and n=6, with an indication of implementation details for systems of larger dimension. The theory is illustrated by application to four differential eigenvalue problems on unbounded intervals: hydrodynamic stablity of boundary-layer flow past a compliant surface, the eigenvalue problem associated with the stability of solitary waves, the stability of Bickley jet in oceanography, and the eigenvalue problem associated with the stability of the Ekman layer in atmospheric dynamics. Received February 2, 2001 / Revised version received May 28, 2001 / Published online October 17, 2001     

An F. and M. Riesz theorem for planar vector fields

We prove that solutions of the homogeneous equation Lu=0, where L is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if is an open subset of the plane with smooth boundary, satisfiesLu=0 on , has tempered growth at the boundary, and its weak boundary value is a measure , then is absolutely continuous with respect to Lebesgue measure on the noncharacteristic portion of . Received March 10, 2000 / Published online April 12, 2001     

A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

Summary A system ofN particles inR ^d with mean field interaction and diffusion is considered. Assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by Brownian sheets (microscopic equation), where all coefficients depend on the position of the particles and on the empirical mass distribution process. This empirical mass distribution process satisfies a quasilinear stochastic partial differential equation (SPDE). This SPDE (mezoscopic equation) is solved for general measure valued initial conditions by extending the empirical mass distribution process from point measure valued initial conditions with total mass conservation. Starting with measures with densities inL ₂(R ^d ,dr), wheredr is the Lebesgue measure, the solution will have densities inL ₂(R ^d ,dr) and strong uniqueness (in the Itô sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDE's.This research was supported by NSF grant DMS92-11438 and ONR grant N00014-91J-1386     

Gauß-Manin determinants for rank 1 irregular connections on curves

Let be a smooth open curve over a field , where k is an algebraically closed field of characteristic 0. Let be a (possibly irregular) absolutely integrable connection on a line bundle L. A formula is given for the determinant of de Rham cohomology with its Gau?-Manin connection . The formula is expressed as a norm from the curve of a cocycle with values in a complex defining algebraic differential characters [7], and this cocycle is shown to exist for connections of arbitrary rank. Received: 13 September 1999 / Published online: 17 August 2001     

Dynamics near extinction time of a singular diffusion equation

We will show that if u is the solution of the equation , in is an even function on and is monotone decreasing in on , , where is a monotone increasing function satisfying with being given by and , then the rescaled function , will converge uniformly on every compact subset of to as where . Received: 25 May 2000 / Revised version: 26 October 2001 / Published online: 28 February 2002     

Partial regularity for variational integrals with (s,\mu ,q)–Growth

We introduce integrands of –type, which are, roughly speaking, of lower (upper) growth rate ) satisfying in addition for some . Then, if , we prove partial –regularity of local minimizers by the way including integrands f being controlled by some N–function and also integrands of anisotropic power growth. Moreover, we extend the known results up to a certain limit and present examples which are not covered by the standard theory. Received: 17 February 2000 / Accepted: 23 January 2001 / Published online: 4 May 2001     

Remarks on the uniqueness of comparable renormalized solutions of elliptic equations with measure data

We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem -div(a(x,Du))=μ in Ω, u=0 on ∂Ω, where μ is a Radon measure with bounded variation on Ω. Received: December 27, 2000 Published online: December 19, 2001     

Ginzburg-Landau equation with magnetic effect in a thin domain

We study the Ginzburg-Landau equation with magnetic effect in a thin domain in , where the thickness of the domain is controlled by a parameter . This equation is an Euler equation of a free energy functional and it has trivial solutions that are minimizers of the functional. In this article we look for a nontrivial stable solution to the equation, that is, a local minimizer of the energy functional. To prove the existence of such a stable solution in , we consider a reduced problem as and a nondegenerate stable solution to the reduced equation. Applying the standard variational argument, we show that there exists a stable solution in near the solution to the reduced equation if is sufficiently small. We also present a specific example of a domain which allows a stable vortex solution, that is, a stable solution with zeros. Received: 11 May 2001 / Accepted: 11 July 2001 /Published online: 19 October 2001     

Hessian equations on compact non-negatively curved riemannian manifolds

On a compact connected riemannian manifold, we partly extend to hessian equations an existence result proved for Monge-Ampère equations. Non-negative curvature is required for a priori estimates. Received: 13 July 2001 / Accepted: 25 October 2001 / Published online: 29 April 2002 The author is supported by the CNRS     

A free boundary problem for $\infty$–Laplace equation

We consider a free boundary problem for the p-Laplacian describing nonlinear potential flow past a convex profile K with prescribed pressure on the free stream line. The main purpose of this paper is to study the limit as of the classical solutions of the problem above, existing under certain convexity assumptions on a(x). We show, as one can expect, that the limit solves the corresponding potential flow problem for the -Laplacian in a certain weak sense, strong enough however, to guarantee uniqueness. We show also that in the special case the limit is given by the distance function. Received: 10 October 2000 / Accepted: 23 February 2001 / Published online: 19 October 2001     

Partial regularity in non-linear elasticity

We prove partial regularity of minimizers of some polyconvex functionals. In particular our results include models such as ∫_Ω a(x,u)(|Du|²+| det Du|²), where a is a bounded H?lder continuous function, such that a(x,u)≥c for some positive constant c. Received: 2 January 2001 / Revised version: 30 August 2001     

Homogenization in general periodically perforated domains by a spectral approach

In this article we study the asymptotic behaviour as tends to 0 of the Neumann problem $-\Delta u_\epsilon+u_\epsilon=\epsilon$-periodic bounded open set of . The period cell of is equal to where is a regular open subset of the d-dimensional torus. We prove that if there exists a smallest integer such that the n-th non-zero eigenvalue of the spectral problem in satisfies , the limiting problem is a linear system of second order p.d.e.'s, of size n. By this spectral approach we extend in the periodic framework a result due to Khruslov without making strong geometrical assumptions on the perforated domain . Received: 20 December 2000 / Accepted: 11 May 2001 / Published online: 19 October 2001     

On interfaces between incompatible wells and elastic deformations of infinite cylinders

We consider the minimization problem for the functional where is an infinitely long cylinder. The density is polyconvex and assumed to be 0 on a set of wells and positive elsewhere. We show that the gradients of solutions with finite energy have to approach one component for and one component for , if the number of components is finite (among other conditions). Moreover, for certain pairs of distinct components we construct nontrivial minimizers within the class of solutions approaching the given components. We follow ideas developed in the variational study of heteroclinic connections for Lagrangian systems and we put special emphasis on multiplicity of such interface solutions. We discuss an application in the theory of nonlinear elasticity, where such solutions are called semi-necks. When a two-dimensional infinite hyperelastic strip is stretched along its infinite direction it may occur that for a given tensile load many homogeneous deformations are possible. In such a case we show by infimizing the energy functional the existence of configurations that tend asymptotically to two different homogeneous deformations. Received: 1 March 2000 / Accepted: 4 December 2000 / Published online: 4 May 2001     

Uniqueness of positive multi-lump bound states of nonlinear Schrödinger equations

In this paper we are concerned with multi-lump bound states of the nonlinear Schr?dinger equation for sufficiently small , where for and for . V is bounded on . For any finite collection of nondegenerate critical points of V, we show the uniqueness of solutions of the form for , where u is positive on and is a small perturbation of a sum of one-lump solutions concentrated near , respectively for sufficiently small . Received: 30 October 2001; in final form: 10 June 2002 /Published online: 2 December 2002 RID="*" ID="*" Research supported by Alexander von Humboldt Foundation in Germany and NSFC in China     

A singular limit arising in combustion theory: Fine properties of the free boundary

The equation where converges to the Dirac measure concentrated at with mass has been used as a model for the propagation of flames with high activation energy. For initial data that are bounded in and have a uniformly bounded support, we study non-negative solutions of the Cauchy problem in as We show that each limit of is a solution of the free boundary problem in on (in the sense of domain variations and in a more precise sense). For a.e. time t the graph of u(t) has a unique tangent cone at -a.e. The free boundary is up to a set of vanishing measure the sum of a countably n-1-rectifiable set and of the set on which vanishes in the mean. The non-degenerate singular set is for a.e. time a countably n-1-rectifiable set. As key tools we introduce a monotonicity formula and, on the singular set, an estimate for the parabolic mean frequency. Received: 8 August 2001 / Accepted: 8 May 2002 / Published online: 5 September 2002 RID="a" ID="a" Partially supported by a Grant-in-Aid for Scientific Research, Ministry of Education, Japan.     

Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d≥3

We consider an asymmetric exclusion process in dimension d≥ 3 under diffusive rescaling starting from the Bernoulli product measure with density 0 < α < 1. We prove that the density fluctuation field Y ^N _t converges to a generalized Ornstein–Uhlenbeck process, which is formally the solution of the stochastic differential equatin dY _t = ?Y _t dt + dB ^∇ _t , where ? is a second order differential operator and B ^∇ _t is a mean zero Gaussian field with known covariances. Received: 31 May 1999 / Revised version: 15 June 2000 / Published online: 24 January 2001     

On the dimensions of automorphism groups of eight-dimensional ternary fields,II

LetT be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed Lie subgroup of its automorphism group which is taken with the compact-open topology. It is proved that if the ternary fixed fieldF of is connected, then is either isomorphic to one of the compact Lie groupsG ₂ or SU₃, or the (covering) dimension of is at most 7.     

Concentration of low energy extremals: Identification of concentration points

We study the variational problem where , is a bounded domain, , F satisfies $0\leq F|t|\leq \alpha |t|^{2^*}$ and is upper semicontinuous. We show that to second order in the value only depends on two ingredients. The geometry of enters through the Robin function (the regular part of the Green's function) and F enters through a quantity which is computed from (radial) maximizers of the problem in . The asymptotic expansion becomes Using this we deduce that a subsequence of (almost) maximizers of must concentrate at a harmonic center of : i.e., , where is a minimum point of . Received: 24 January 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001     

The local Rankin-Selberg convolution for GL(n): divisibility of the conductor

Let F be a non-Archimedean local field and an integer. Let be irreducible supercuspidal representations of GL with . One knows that there exists an irreducible supercuspidal representation of GL, with , such that the local constants (in the sense of Jacquet, Piatetskii-Shapiro and Shalika) are distinct. In this paper, we show that, when is an unramified twist of , one may here takem dividingn and , for a prime divisor ofn depending on and the order of : in particular, , where is the least prime divisor of . This follows from a result giving control of certain divisibility properties of the conductor of a pair of supercuspidal representations. Received: 11 November 2000 / Accepted: 15 January 2001 / Published online: 23 July 2001