Differential equations with singular fields

This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the vector field, a compressibility condition on the flow (bounded Jacobian) is considered. The main result provides existence under the condition that the vector field belongs to BV in dimension 2 and SBV in higher dimensions.     

General existence results for reflected BSDE and BSDE

In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z. We deal with the case of a bounded terminal condition ξ and a bounded barrier L as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in Essaky and Hassani (submitted for publication) [14]. The work is suggested by the interest the results might have in finance, control and game theory.     

Conformal fields and the stability of leaves with constant higher order mean curvature

In this paper, we study hypersurfaces with constant rth mean curvature Sr. We investigate the stability of such hypersurfaces in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros and Sousa, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is an rth Jacobi field of a hypersurface with Sr+1 constant. Finally, we study relations between rth Jacobi fields and vector fields preserving a foliation.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

Dimensional reduction for energies with linear growth involving the bending moment

A Γ-convergence analysis is used to perform a 3D–2D dimension reduction of variational problems with linear growth. The adopted scaling gives rise to a nonlinear membrane model which, because of the presence of higher order external loadings inducing a bending moment, may depend on the average in the transverse direction of a Cosserat vector field, as well as on the deformation of the mid-plane. The assumption of linear growth on the energy leads to an asymptotic analysis in the spaces of measures and of functions with bounded variation.     

Vector Variational Inequalities and the (S)<Subscript>+</Subscript> Condition

Let $${\cal Z}$$ and X be Hausdorff real topological vector spaces and let $${\cal L}_b(X,{\cal Z})$$ be the space of continuous linear mappings from X into $${\cal Z}$$ equipped with the topology of bounded convergence. In this paper, we define the (S)₊ condition for operators from a nonempty subset of X into $${\cal L}_b(X,{\cal Z})$$ and derive some existence results for vector variational inequalities with operators of the class (S)₊. Some applications to vector complementarity problems are given.     

The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds

Let Ω be a bounded domain with C²-smooth boundary in an n-dimensional oriented Riemannian manifold. It is well known that for the biharmonic equation Δ²u=0 in Ω with the condition u=0 on ∂Ω, there exists an infinite set {uk} of biharmonic functions in Ω with positive eigenvalues {λk} satisfying on ∂Ω. In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of the biharmonic Steklov eigenvalues λk.     

Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition (λ₁,…,λd²) of size ?d there exists k?1 such that the tensor square of the irreducible representation of the symmetric group Sk?d with respect to the rectangular partition (k?,…,k?) contains the irreducible representation corresponding to the stretched partition (kλ₁,…,kλd²). We also prove a related approximate version of this statement in which the stretching factor k is effectively bounded in terms of d. We further discuss the consequences for geometric complexity theory which provided the motivation for this work.     

On duality and fractionality of multicommodity flows in directed networks

In this paper we address a topological approach to multiflow (multicommodity flow) problems in directed networks. Given a terminal weight μ, we define a metrized polyhedral complex, called the directed tight span T_μ, and prove that the dual of the μ-weighted maximum multiflow problem reduces to a facility location problem on T_μ. Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by μ. By utilizing this duality, we establish the classifications of terminal weights admitting a combinatorial min–max relation (i) for every network and (ii) for every Eulerian network. Our result includes the Lomonosov–Frank theorem for directed free multiflows and Ibaraki–Karzanov–Nagamochi’s directed multiflow locking theorem as special cases.     

Strong trajectory attractors for dissipative Euler equations

The 2D Euler equations with periodic boundary conditions and extra linear dissipative term Ru, R>0 are considered and the existence of a strong trajectory attractor in the space is established under the assumption that the external forces have bounded vorticity. This result is obtained by proving that any solution belonging the proper weak trajectory attractor has a bounded vorticity which implies its uniqueness (due to the Yudovich theorem) and allows to verify the validity of the energy equality on the weak attractor. The convergence to the attractor in the strong topology is then proved via the energy method.     

Embedding polynomial matrices of one variable

A non square matrix with coefficients in K[z] can (if a condition on its minors is satisfied) be embedded into a square matrix with determinant 1. Finding theoretically and in an algorithmic way an embedding of small degree is solved by a construction with vector bundles on the projective line over K.     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

On the search for weighted inequalities for operators related to the Ornstein-Uhlenbeck semigroup

In this paper, for each given $ we characterize the weights v for which the centered maximal function with respect to the gaussian measure and the Ornstein-Uhlenbeck maximal operator are well defined for every function in and their means converge almost everywhere. In doing so, we find that this condition is also necessary and sufficient for the existence of a weight u such that the operators are bounded from into We approach the poblem by proving some vector valued inequalities. As a byproduct we obtain the strong type (1,1) for the “global” part of the centered maximal function. Received May 18, 1999 / Revised December 9, 1999 Published online July 20, 2000     

Limit cycles, invariant meridians and parallels for polynomial vector fields on the torus

We study the polynomial vector fields of arbitrary degree in R³ having the 2-dimensional torus invariant by their flow. We characterize all the possible configurations of invariant meridians and parallels that these vector fields can exhibit. Furthermore we analyze when these invariant either meridians or parallels can be limit cycles.     

Anti-self-dual connections and their related flow on 4-manifolds

In this paper, we establish the existence of the maximal time for a smooth solution to the anti-self-dual (ASD) flow in vector bundles over a 4-dimensional compact Riemannian manifold M and present a different proof of the Taubes’ existence theorem on anti-self-dual connections on 4-manifolds.     

The <Emphasis Type="Italic">(S)</Emphasis><Stack><Subscript>+</Subscript><Superscript>1</Superscript></Stack>Condition for Generalized Vector Variational Inequalities

In this paper, we extend the (S) ₊ ¹ condition to multivalued mappings in an ordered Hausdorff topological vector space and we derive some existence results for generalized vector variational inequalities associated with multivalued mappings satisfying the (S) ₊ ¹ condition. We generalize also an existence result of Cubiotti and Yao for generalized variational inequalities of class (S) ₊ ¹ to barreled normed spaces. As consequences, some existence results for vector variational inequalities are established.This work was partially supported by grants from the National Science Council of the Republic of China. Communicated by H. P. Benson     

Backward iteration in strongly convex domains

We prove that a backward orbit with bounded Kobayashi step for a hyperbolic, parabolic or strongly elliptic holomorphic self-map of a bounded strongly convex C² domain in Cd necessarily converges to a repelling or parabolic boundary fixed point, generalizing previous results obtained by Poggi-Corradini in the unit disk and by Ostapyuk in the unit ball of Cd.     

Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Given a finite root system Φ, we show that there is an integer c=c(Φ) such that , for any reductive algebraic group G with root system Φ and any irreducible rational G-modules L, L^′. There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Extn, for any integer n?0, using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L=L(λ), even allowing λ to vary, provided the p-adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.     

Periodic orbits for a class of C 1 three-dimensional systems

We studyC ¹ perturbations of a reversible polynomial differential system of degree 4 in. We introduce the concept of strongly reversible vector field. If the perturbation is strongly reversible, the dynamics of the perturbed system does not change. For non-strongly reversible perturbations we prove the existence of an arbitrary number of symmetric periodic orbits. Additionally, we provide a polynomial vector field of degree 4 in with infinitely many limit cycles in a bounded domain if a generic assumption is satisfied. The first two authors are partially supported by a MCYT grant number MTM2005-06098-C02-01, and by a CICYT grant number 2005SGR 00550. The second author is partially supported by a FAPESP-BRAZIL grant 10246-2. All authors are also supported by the joint project CAPES-MECD grant HBP2003-0017.     

Ahlfors-régularité des quasi-minima de Mumford–Shah

Let be an open set. We consider on Ω the competitors (U,K) for the reduced Mumford–Shah functional, that is to say the Mumford–Shah functional in which the -norm of U term is removed, where K is a closed subset of Ω and U is a function on ΩK with gradient in  . The main result of this paper is the following: there exists a constant c for which, whenever (U,K) is a quasi-minimizer for the reduced Mumford–Shah functional and B(x,r) is a ball centered on K and contained in Ω with bounded radius, the -measure of is bounded above by cr^N−1 and bounded below by c⁻¹r^N−1.