Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in R^m,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0. R. H.’s research partially supported by the National Science Foundation.     

Singular cycles of vector fields on regular parts of the boundary of Morse-Smale systems

At the boundary of the class of Morse-Smale vector fields there are vector fields whose unique degenerate phenomena is a singular cycle. We first characterize and classify all singular cycles which contains only one degeneracy (thesimple singular cycles: ssc). Each of these cycles defines a condimension one submanifold of vector fields. For some ssc its codimension one submanifold is a regular part of the boundary of the Morse-Smale systems. We characterize those ssc that defines this type of submanifold. Our ambient space isn dimensional,n2.Supported by Fondecyt, Proyecto 1930863.     

On a class of some special sets on thek-skeleton of a convex compact set

In this paper, generalizing the notion of a path we define ak-area to be the setD={g(t):t ∈J} on thek-skeleton of a convex compact setK in a Hilbert space, whereg is a continuous injection map from thek-dimensional convex compact setJ to thek-skeleton ofK. We also define anE ^k-area onK, whereE ^k is ak-dimensional subspace, to be ak-area with the propertyπ(g(t))=t,t ∈π(K), whereπ is the orthogonal projection onE ^k. This definition generalizes the notion of an increasing path on the 1-skeleton ofK. The existence of such sets is studied whenK is a subset of a Euclidean space or of a Hilbert space. Finally some conjectures are quoted for the number of such sets in some special cases.     

The propagation of singularities along gliding rays

Summary LetP be a second-order differential operator with real principal symbol and fibre-simple characteristics on a manifold with boundary non-characteristic forP. LetB be a differential operator such that the boundary value problem (P, B) is normal and satisfies the Lopatinskii-Schapiro condition. The singularities of distributions,u, such thatP u is smooth on the boundary, near points at which the boundary is bicharacteristically convex are shown to propagate, in the boundary, only along the gliding rays, which are the leaves of the Hamilton foliation of the glancing surface. This analysis, combined with known results on diffraction, leads to a Poisson relation bounding the singular support of the Fourier transform of the Dirichlet spectral density for a compact Riemannian manifold with geodesically convex, or concave, boundary in terms of the geodesic length spectrum.     

On surfaces and their contours

We prove two theorems concerning the global behaviour of a smooth compact surfaceS, without boundary, embedded in a real projective space or mapped to a plane. Our starting point is an analysis of the orientability properties of the normal bundle of a singular projective curve. Then we see how an excellent projection fromS to the Euclidean plane gives rise to integral relations linking the singularities of the apparent contour. Finally, given an embedding ofS in RPⁿ, we look at the discriminant Δ^* of a net of hyperplanes that intersectsS in a generic way, obtaining a characterization of Δ^* in terms of mod.2 cohomology invariants.     

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍ n

In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ ⁿ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍ ⁿ .We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones. Our main result describes which are the CMC hypersurfaces of revolution in ℍ ⁿ .The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ ⁿ .     

Basic intersection cohomology of conical fibrations

We introduce notions of singular fibration and singular Seifert fibration. These notions naturally generalize that o locally trivial fibration to the category of stratified pseudomanifolds. For singular foliations determined by such fibrations, we prove the de Rham theorem for basic intersection cohomology recently introduced by the present authors. One of the main examples of such a structure is the natural projection to the space of fibers of a singular Riemannian foliation determined by a Lie group action on a compact smooth manifold.     

Controllability properties of constrained linear systems

In this paper, we present results on constrained controllability for linear control systems. The controls are constrained to take values in a compact set containing the origin. We use the results on reachability properties discussed in Ref. 1.We prove that controllability of an arbitrary pointp inR ⁿ is equivalent to an inclusion property of the reachable sets at certain positive times. We also develop geometric properties ofG, the set of all nonnegative times at whichp is controllable, and ofC, the set of all controllable points. We characterize the setC for the given system and provide additional spectrum-dependent structure.We show that, for the given linear system, several notions of constrained controllability of the pointp are the same, and thus the setC is open. We also provide a necessary condition for small-time (differential or local) constrained controllability ofp.This work was supported in part by NSF Grant ECS-86-09586.     

Paley-Wiener theorem for singular support of K-finite distributions on symmetric spaces

We study Fourier transforms of distributions on a symmetric space X. Eguchi et al. [1] characterized the image of $\text{E}$′(X)-distributions of compact support under the Fourier transform. We give a simpler proof of Eguchi's result and characterize the size of the singular support for the K-finite members of E′(X). We apply this Paley-Wiener type theorem to invariant differential equations on X.     

Rotationally symmetric 1-harmonic maps from D 2 to S 2

We consider rotationally symmetric 1-harmonic maps from D ² to S ² subject to Dirichlet boundary conditions. We prove that the corresponding energy—a degenerate non-convex functional with linear growth—admits a unique minimizer, and that the minimizer is smooth in the bulk and continuously differentiable up to the boundary. We also show that, in contrast with 2-harmonic maps, a range of boundary data exists such that the energy admits more than one smooth critical point: more precisely, we prove that the corresponding Euler–Lagrange equation admits a unique (up to scaling and symmetries) global solution, which turns out to be oscillating, and we characterize the minimizer and the smooth critical points of the energy as the monotone, respectively non-monotone, branches of such solution. R. Dal Passo passed away on 8th August 2007. Endowed with great strength, creativity and humanity, Roberta has been an outstanding mathematician, an extraordinary teacher and a wonderful friend. Farewell, Roberta.     

On the dynamics of predator-prey models with the Beddington–De Angelis functional response, under Robin boundary conditions

The paper concerns the Beddington–De Angelis predator-prey model, under Robin boundary conditions. General properties—such as boundedness, uniqueness and existence of invariant regions—are obtained. Linear stability (instability) threshold of the equilibrium state S (biologically meaningful) and diffusion-driven instability (Turing effect) are studied. In the framework of nonlinear L ²-energy stability, conditions guaranteeing stability and local attractiveness are obtained.      

Basic intersection cohomology of conical fibrations

We introduce notions of singular fibration and singular Seifert fibration. These notions naturally generalize that o locally trivial fibration to the category of stratified pseudomanifolds. For singular foliations determined by such fibrations, we prove the de Rham theorem for basic intersection cohomology recently introduced by the present authors. One of the main examples of such a structure is the natural projection to the space of fibers of a singular Riemannian foliation determined by a Lie group action on a compact smooth manifold.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 235–257.Original Russian Text Copyright © 2005 by M. Saralegi-Aranguren, R. Wolak.This revised version was published online in April 2005 with a corrected issue number.     

E-varieties of regular semigroups,relatively bifree objects and fully invariant congruences

Analogous to the concept of a free object on a setX in a variety of algebras is the notion of a bifree object onX in an e-variety of regular semigroups. If an e-variety contains a bifree object onX, then a homomorphic image of that bifree object is itself bifree onX in some e-variety if and only if the corresponding congruence is fully invariant. Furthermore, the lattice of e-subvarieties of any locally inverse or E-solid e-variety ε is antiisomorphic with the lattice of all fully invariant congruences on the bifree object on a countably infinite setX in ε. We give a Birkhoff-type theorem for classes of locally inverse or E-solid semigroups, and we give an intrinsic test for whether or not a regular semigroup is bifree onX in the e-variety it generates.     

Necessary optimality conditions for a singular surface in the form of synthesis

For a linear control problem using the traditional open-loop approach, a new representation for the singular control and generalized, invariant conditions for optimality are found. The phase portrait of a nonlinear control problem is considered in the neighborhood of singular trajectories. The singular paths form a hypersurface, approached by regular paths from both sides. The Bellman function for this problem is a classical (smooth) solution to a first-order PDE with nonsmooth Hamiltonian over two smooth (regular) branches, related to the halfneighborhoods of the surface. These solutions are at least twice differentiable and have first discontinuous derivatives of odd order. The invariant form for these necessary conditions is found in terms of Jacobi (Poisson) brackets, consisting of several equalities and inequalities. The latter relations guarantee the validity of the Kelley condition as well as the geometrical constraints for the singular control variables. Thus, the Kelley condition appears to be just a certain property of a smooth solution to a first-order PDE with nonsmooth Hamiltonian. All the relations, including the Hamiltonian equations of singular motion, do not use singular controls; they are based on regular Hamiltonians depending only upon the state vector and the gradient of the Bellman function (adjoint vector).This work was suported by Grant No. 93-013-16285 of the Russian Fund for Fundamental Research.     

On a class of nonlocal parabolic equations

We consider a boundary value problem for parabolic equations with nonlocal nonlinearity of such a form that favorably differs from other equations in that it leads to partial differential equations that have important properties of ordinary differential equations. Local solvability and uniqueness theorems are proved, and an analog of the Painlevé singular nonfixed points theorem is proved. In this case, there is an alternative—either a solution exists for all t ≥ 0 or it goes to infinity in a finite time t = T (blowup mode). Sufficient conditions for the existence of a blowup mode are given.     

On the solution of Maxwell's equations in polygonal domains

This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H² when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H², and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline

In this paper, we present B-spline method for numerically solving singular two-point boundary value problems for certain ordinary differential equation having singular coefficients.These problems arise when reducing partial differential equation to ordinary differential equation by physical symmetry. To remove the singularity, we first use Chebyshev economizition in the vicinity of the singular point and the boundary condition at a point x=δ (in the vicinity of the singularity) is derived. The resulting regular BVP is then efficiently treated by employing B-spline for finding the numerical solution. Some examples have been included and comparison of the numerical results made with other methods.     

Partial regularity of energy minimizing harmonic maps into a complete manifold

In this paper, we study energy minimizing harmonic maps into a complete Riemannian manifold. We prove that the singular set of such a map has Hausdorff dimension at mostn–2, wheren is the dimension of the domain. We will also give an example of an energy minimizing map from surface to surface that has a singular point. Thus then–2 dimension estimate is optimal, in contrast to then–3 dimension estimate of Schoen-Uhlenbeck [SU] for compact targets.     

A viability theorem for set-valued states in a Hilbert space

Applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact (possibly non-convex) sets rather than vectors. This leads to evolutions in a basic set which can be supplied with a metric (like the well-established Pompeiu–Hausdorff distance), but it does not have an obvious linear structure. This article extends differential inclusions with state constraints to compact-valued states in a separable Hilbert space H. The focus is on sufficient conditions such that a given constraint set (of compact subsets) is viable a.k.a. weakly invariant. Our main result extends the tangential criterion in the well-known viability theorem (usually for differential inclusions in a vector space) to the metric space of non-empty compact subsets of H.