Local minimizers in micromagnetics and related problems

Let be a smooth bounded domain and consider the energy functional Here is a small parameter and the admissible function m lies in the Sobolev space of vector-valued functions and satisfies the pointwise constraint for a.e. . The induced magnetic field is related to m via Maxwell's equations and the function is assumed to be a sufficiently smooth, non-negative energy density with a multi-well structure. Finally is a constant vector. The energy functional arises from the continuum model for ferromagnetic materials known as micromagnetics developed by W.F. Brown [9]. In this paper we aim to construct local energy minimizers for this functional. Our approach is based on studying the corresponding Euler-Lagrange equation and proving a local existence result for this equation around a fixed constant solution. Our main device for doing so is a suitable version of the implicit function theorem. We then show that these solutions are local minimizers of in appropriate topologies by use of certain sufficiency theorems for local minimizers. Our analysis is applicable to a much broader class of functionals than the ones introduced above and on the way to proving our main results we reflect on some related problems. Received: 20 November 2000 / Accepted: 4 December 2000 / Published online: 4 May 2001     

Triangular Witt groups Part II: From usual to derived

   Abstract. We establish that the derived Witt group is isomorphic to the usual Witt group when 2 is invertible. This key result opens the Ali Baba's cave of triangular Witt groups, linking the abstract results of Part I to classical questions for the usual Witt group. For commercial purposes, we survey the future applications of triangular Witt groups in the introduction. We also establish a connection between odd-indexed Witt groups and formations. Finally, we prove that over a commutative local ring in which 2 is a unit, the shifted derived Witt groups are all zero but the usual one. Received July 15, 1999; in final form November 8, 1999 / Published online October 30, 2000     

A note on trilinear forms for reducible representations and Beilinson’s conjectures

We extend Prasad’s results on the existence of trilinear forms on representations of GL ₂ of a local field, by permitting one or more of the representations to be reducible principal series, with infinite-dimensional irreducible quotient. We apply this in a global setting to compute (unconditionally) the dimensions of the subspaces of motivic cohomology of the product of two modular curves constructed by Beilinson. Received February 24, 2000 / final version received September 12, 2000?Published online November 8, 2000     

The multichoice consistent value

We consider multichoice NTU games, i.e., cooperative NTU games in which players can participate in the game with several levels of activity. For these games, we define and characterize axiomatically the multichoice consistent value, which is a generalization of the consistent NTU value for NTU games and of the multichoice value for multichoice TU games. Moreover, we show that this value coincides with the consistent NTU value of a replicated NTU game and we provide a probabilistic interpretation. Received: May 1998/Final version: January 2000     

The local root number of elliptic curves with wild ramification

Let E be an elliptic curve over a number field F. The root number is conjecturally the sign of the functional equation of L-function of E/F. It is defined as the product of local signs over all places of F. The purpose of this paper is to describe this local sign by the coefficients of a Weierstra? equation of E. Received: 31 March 2000 / Revised version: 22 November 2001 / Published online: 23 May 2002     

The genealogy of continuous-state branching processes with immigration

Recent works by J.F. Le Gall and Y. Le Jan [15] have extended the genealogical structure of Galton-Watson processes to continuous-state branching processes (CB). We are here interested in processes with immigration (CBI). The height process H which contains all the information about this genealogical structure is defined as a simple local time functional of a strong Markov process X ^*, called the genealogy-coding process (GCP). We first show its existence using It?’s synthesis theorem. We then give a pathwise construction of X ^* based on a Lévy process X with no negative jumps that does not drift to +∞ and whose Laplace exponent coincides with the branching mechanism, and an independent subordinator Y whose Laplace exponent coincides with the mechanism. We conclude the construction with proving that the local time process of H is a CBI-process. As an application, we derive the analogue of the classical Ray–Knight–Williams theorem for a general Lévy process with no negative jumps. Received: 28 January 2000 / Revised version: 5 February 2001 / Published online: 11 December 2001     

Change of generalized indices of stationary solutions. to multiparametric optimization

We study the local change of the generalized index, which is a modification of the Morse index and the stationary index, for the multiparametric optimization. Under the Regular Value Condition, the change of the generalized index around a triplet (x,v,t) is locally bounded by the dimension of the parameter vector t, where x is a variable vector and v a vector of the Lagrange multiplier space. We also discuss the local change of the generalized index around a pair (x,t). Received: March 27, 1998 / Accepted: January 29, 2000?Published online April 20, 2000     

On the minimum problem for nonconvex, multiple integrals of product type

We consider the problem of minimizing multiple integrals of product type, i.e. where is a bounded, open set in , is a possibly nonconvex, lower semicontinuous function with p-growth at infinity for some and the boundary datum is in (or simply in if ). Assuming that the convex envelope off is affine on each connected component of the set , we prove attainment for () for every continuous, positively bounded below function g such that (i) every point is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that the class of coefficents g that yield existence to () is dense in the space of continuous, positive functions on . We present examples which show that these conditions for attainment are essentially sharp. Received April 12, 2000 / Accepted May 9, 2000 / Published online November 9, 2000     

Existence, multiplicity and concentration of bound states for a quasilinear elliptic field equation

In this paper we study existence, multiplicity and concentration of solutions for the following nonlinear field equation where the potential V is positive and W is an appropriate singular function. Here is regarded as a small parameter. Under suitable conditions on V and W we find solutions exhibiting a concentration behaviour at an absolute minimum of V as Such solutions are obtained as local minima for the associated functional; the proofs of our results rely on a careful analysis of the behaviour of minimizing sequences and use arguments inspired by the concentration-compactness principle. Received July 21, 1999; Accepted April 9, 2000 / Published online September 14, 2000     

A linear boundary value problem for weakly quasiregular mappings in space

Given a number a weakly L-quasiregular map on a domain in space is a map u in a Sobolev space that satisfies almost everywhere in In this paper, we study the problem concerning linear boundary values of weakly L-quasiregular mappings in space with dimension It turns out this problem depends on the power p of the underlying Sobolev space. For p not too far below the dimension n we show that a weakly quasiregular map in can only assume a quasiregular linear boundary value; however, for all and , we prove a rather surprising existence result that every linear map can be the boundary value of a weakly L-quasiregular map in Received July 20, 2000 / Accepted September 22, 2000 / Published online December 8, 2000     

Constant mean curvature surfaces of annular type

We consider large solutions of annular type to the volume constrained Douglas problem. They are conformally immersed H-surfaces. By rescaling we set the volume functional at one while the boundary curves shrink to the origin. We show that the solutions become spherical in a precise manner. Spherical bubbling may fail if the conformality condition is dropped. We also discuss the rotationally symmetric annular solutions to the H-surface equation and consider some illustrative examples. Received: 2 May 2000 / Accepted: 23 January 2001 / Published online: 4 May 2001     

Informativeness of functionals

We introduce the concept of informativeness of a continuous functional on a metric spaceX with respect to a setMX and a metric x. We pose the problem of finding the most informative functional. For some sets of continuous functions, this problem is solved by reduction to a subset of functionals given by the value of a function at a certain point.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1156–1163, September, 1994.     

On the second derivative of the spherical contact distribution function of smooth grain models

We consider a stationary grain model Ξ in ℝ ^d with convex, compact and smoothly bounded grains. We study the spherical contact distribution function F of Ξ and derive (under suitable assumptions) an explicit formula for its second derivative F″. The value F″(0) is of a simple form and admits a clear geometric interpretation.For the Boolean model we obtain an interesting new formula for the(d− 1)-st quermass density. Received: 22 November 1999 / Revised version: 2 November 2000 /?Published online: 14 June 2001     

An additive Schwarz preconditioner for the FETI method

Summary.  A new additive Schwarz preconditioner for the Finite Element Tearing and Interconnecting (FETI) method is analyzed in this paper. This preconditioner has the unique feature that the coefficient matrix of its ``coarse grid' problem is mesh independent. For a model second order heterogeneous elliptic boundary value problem in two dimensions, the condition number of the preconditioned system is shown to be bounded by C[1+ln(H/h)]², where h is the mesh size, H is the typical diameter of the subdomains, and the constant C is independent of h, H, the number of subdomains and the coefficients of the boundary value problem. Received May 8, 2000 / Revised version received January 2, 2002 / Published online July 18, 2002 Mathematics Subject Classification (1991): 65N55, 65N30     

Numerical approximation of entropy solutions for hyperbolic integro-differential equations

Summary. Scalar hyperbolic integro-differential equations arise as models for e.g. radiating or self-gravitating fluid flow. We present finite volume schemes on unstructured grids applied to the Cauchy problem for such equations. For a rather general class of integral operators we show convergence of the approximate solutions to a possibly discontinuous entropy solution of the problem. For a specific model problem in radiative hydrodynamics we introduce a convergent fully discrete finite volume scheme. Under the assumption of sufficiently fast spatial decay of the entropy solution we can even establish the convergence rate h^1/4|ln(h)| where h denotes the grid parameter. The convergence proofs rely on appropriate variants of the classical Kruzhkov method for local balance laws together with a truncation technique to cope with the nonlocal character of the integral operator.Mathematics Subject Classification (2000): 35L65, 35Q35, 65M15     

C*-algebras of singular integral operators in domains with oscillating conical singularities

 For a domain with singular points on the boundary, we consider a C ^* -algebra of operators acting in the weighted space . It is generated by the operators of multiplication by continuous functions on and the operators where σ is a homogeneous function. We show that the techniques of limit operators apply to define a symbol algebra for . When combined with the local principle, this leads to describing the Fredholm operators in . Received: 21 December 2000     

Closed stability index of excellent henselian local rings

We show that the closed stability index of an excellent henselian local ring of real dimension d2 with real closed residue field is When d=2 it is shown that the value of can be either 2 or 3 and give characterizations of each of these values in terms of the relation of A with its normalization and in terms of the real spectrum of A.Mathematics Subject Classification (2000): 14P15, 32B10, 13J15, 13J25Partially supported by DGES BFM2002-04797 and EC contract HPRN-CT-2001-00271in final form: 15 October 2003     

The Yang-Mills measure in the Kauffman bracket skein module

For each closed, orientable surface , we construct a local, diffeomorphism invariant trace on the Kauffman bracket skein module . The trace is defined when |t| is neither 0 nor 1, and at certain roots of unity. At t = − 1, the trace is integration against the symplectic measure on the SU(2) character variety of the fundamental group of . Received: June 2, 2000     

Complete <Emphasis Type="Italic">k</Emphasis>-Curvature Homogeneous Pseudo-Riemannian Manifolds

Abstract. For k ≥ 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). All the local scalarWeyl invariants of these manifolds vanish. These manifolds are Ricci flat, Osserman, and Ivanov-Petrova. Mathematics Subject Classification (2000): 53B20