Convergence theorems for the practical eigenvector free methods of Gay and Goerisch are obtained under a variety of hypotheses, so that our theorems apply to both traditional boundary-value problems and atomic problems. In addition, we prove convergence of the T*T method of Bazley and Fox without an alignment of projections hypothesis required in previous literature.

By means of a certain variational principle, where rank-one convexity enters in a fundamental way as part of feasibility, we construct a whole class of rank-one convex functions. We show the existence of optimal paths and motivate our analysis by the issue of the equivalence of quasiconvexity and rank-one convexity for 2 × 2 matrices.

This work is concerned with basic structural properties of first-order hyperbolic systems with source terms divided by a small parameter ε. We identify a relaxation criterion necessary for the solution sequences indexed with ε to have reasonable limits as ε goes to zero. This relaxation criterion is shown to imply hyperbolicity of the reduced systems governing the limits. Moreover, we introduce a so-called GC-stability theory and strengthen the hyperbolicity result. The latter shows that there are no linearly stable hyperbolic relaxation approximations for non-hyperbolic conservation laws.

In this paper we study the existence, non-existence and simplicity of the first eigenvalue of the perturbed Hardy-Sobolev operator under various assumptions on the perturbation q. We study the asymptotic behaviour of the first eigenfunction near the origin when the perturbation q is q = s, 0 < s < 1. We will also establish the best constant in a Hardy-Sobolev inequality proved by Adimurthi et al.

The stability of linear initial–boundary-value problems for hyperbolic systems (with constant coefficients) is linked to the zeros of the so-called Lopatinskii determinant. Depending on the location of these zeros, problems may be either unstable, strongly stable or weakly stable. The first two classes are known to be ‘open’, in the sense that the instability or the strong stability persists under a small change of coefficients in the differential operator and/or in the boundary condition.

Here we show that a third open class exists, which we call ‘weakly stable of real type’. Many examples of physical or mathematical interest depend on one or more parameters, and the determination of the stability class as a function of these parameters usually needs an involved computation. We simplify it by characterizing the transitions from one open class to another one. These boundaries are easier to determine since they must solve some overdetermined algebraic system.

Applications to the wave equation, linear elasticity, shock waves and phase boundaries in fluid mechanics are given.

W1,p-gradient Young measures supported in the set Q2(K) of two-dimensional K-quasiconformal matrices are studied. We prove that these Young measures can be generated by gradients of K-quasiregular mappings. This leads, for example, to the 0-1 law for quasiregular W1,p-gradient Young measures and other quasiregular properties such as higher integrability.

We prove that the two-dimensional Brown–Ravenhall operator is bounded from below when the coupling constant is below a specified critical value—a property also referred to as stability. As a consequence, the operator is then self-adjoint. The proof is based on the strategy followed by Evans et al. and Lieb and Yau, with some relevant changes characteristic of the dimension. Our analysis also yields a sharp Kato inequality.

We study a class of quasilinear elliptic problems with diffusion matrices that have at least one diagonal coefficient that blows up for a finite value of the unknown; the other coefficients being continuous with respect to the unknown (without any growth assumption). We introduce two equivalent notions of solutions for such problems and we prove an existence result in these frameworks. Under additional local assumptions on the coefficients, we also establish the uniqueness of the solution. In that case, and when the non-diagonal coefficients are bounded, this unique (generalized) solution is also the unique weak solution strictly less than the value where the diagonal coefficient blows up.

By means of a certain variational principle, where rank-one convexity enters in a fundamental way as part of feasibility, we construct a whole class of rank-one convex functions. We show the existence of optimal paths and motivate our analysis by the issue of the equivalence of quasiconvexity and rank-one convexity for 2 × 2 matrices.

W1,p-gradient Young measures supported in the set Q2(K) of two-dimensional K-quasiconformal matrices are studied. We prove that these Young measures can be generated by gradients of K-quasiregular mappings. This leads, for example, to the 0-1 law for quasiregular W1,p-gradient Young measures and other quasiregular properties such as higher integrability.

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

The notion of gluing of abelian categories was introduced in a paper by Kazhdan and Laumon in 1988 and studied furtherby Polishchuk. We observe that this notion is a particular case of a general categorical construction.

We then apply this general notion to the study of the ring of global differential operators $\mathcal{D}$ on the basic affinespace $G/U$ (here $G$ is a semi-simple simply connected algebraic group over $\mathbb{C}$ and $U\subset G$ is a maximalunipotent subgroup).

We show that the category of $\mathcal{D}$-modules is glued from $|W|$ copies of the category of $D$-modules on $G/U$ where$W$ is the Weyl group, and the Fourier transform is used to define the gluing data. As an application we prove that thealgebra $\mathcal{D}$ is Noetherian, and get some information on its homological properties.

AMS 2000 Mathematics subject classification: Primary 13N10; 16S32; 17B10; 18C20

The connection between the discrete and the continuous coagulation–fragmentation models is investigated. A weak stability principle relying on a priori estimates and weak compactness in L1 is developed for the continuous model. We approximate the continuous model by a sequence of discrete models and, writing the discrete models as modified continuous ones, we prove the convergence of the latter towards the former with the help of the above-mentioned stability principle. Another application of this stability principle is the convergence of an explicit time and size discretization of the continuous coagulation-fragmentation model.

We use singularity theory to classify forced symmetry-breaking bifurcation problems where f1 is 𝕆(2)-equivariant and f2 is 𝔻n-equivariant with the orthogonal group actions on z ∈ ℝ2. Forced symmetry breaking occurs when the symmetry of the equation changes when parameters are varied. We explicitly apply our results to the branching of subharmonic solutions in a model periodic perturbation of an autonomous equation and sketch further applications.

Convergence theorems for the practical eigenvector free methods of Gay and Goerisch are obtained under a variety of hypotheses, so that our theorems apply to both traditional boundary-value problems and atomic problems. In addition, we prove convergence of the T*T method of Bazley and Fox without an alignment of projections hypothesis required in previous literature.

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition:where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

We consider the inverse connection problem consisting of determining a gauge field on $\mathbb{R}^d$ from its non-abelian Radontransform along oriented straight lines. The determination is considered modulo gauge transformations. Our resultsinclude: global uniqueness theorems for $d\geq3$, new local uniqueness theorems for $d=2$, constructive proofs(i.e. proofs containing reconstruction procedures), counterexamples to the global uniqueness for $d=2$, a reduction tothe attenuated X-ray transform.

AMS 2000 Mathematics subject classification: Primary 53C65; 81U40

We introduce the totally multiplicatively prime algebras as those normed algebras for which there exists a positive number K such that K‖F‖‖a‖ ≤ ‖WF,a‖ for all F in M(A) (the multiplication algebra of A) and a in A, where WF,a denotes the operator from M(A) into A defined by WF,a(T) = FT(a) for all T in M(A). These algebras are totally prime and their multiplication algebra is ultraprime. We get the stability of the class of totally multiplicatively prime algebras by taking central closure. We prove that prime H*-algebras are totally multiplicatively prime and that the ℓ1-norm is the only classical norm on the free non-associative algebras for which these are totally multiplicatively prime.

Surfaces arising in amorphous thin-film-growth are often described by certain classes of stochastic PDEs. In this paper we address the question of existence of a solution and statistical quantities (e.g. mean interface width or correlation functions). Moreover, we discuss the approximations of such statistical quantities by the spectral Galerkin method. This is an important question, as the numerical computation of statistical quantities plays a key role in the verification of the models.

We consider a one-dimensional stochastic model of sediment deposition in which the complete time history of sedimentation is the sum of a linear trend and a fractional Brownian motion wH(t) with self-similarity parameter H ∈ (0, 1). The thickness of the sedimentary layer as a function of time, d(t), looks like the Cantor staircase. The Hausdorff dimension of the points of growth of d(t) is found. We obtain the statistical distribution of gaps in the sedimentary record, periods of time during which the sediments have been eroded. These gaps define sedimentary unconformities. In the case H = 1/2 we obtain the statistical distribution of layer thicknesses between unconformities and investigate the multifractality of d(t). We show that the multifractal structures of d(t) and the local time function of Brownian motion are identical; hence d(t) is not a standard multifractal object. It follows that natural statistics based on local estimates of the sedimentation rate produce contradictory estimates of the range of local dimension for d(t). The physical object d(t) is interesting in that it involves the above anomalies, and also in its mechanism of fractality generation, which is different from the traditional multiplicative process.