Trace theorems for vector fields

In the setting of Carnot-Carathéodory spaces we prove some trace theorems for Sobolev functions. We consider the trace on a non characteristic surface for H?rmander vector fields of step and the trace on the boundary of a class of domains in the Grushin plane. Received: 27 January 2000 / in final form: 7 May 2001 / Published online: 1 February 2002     

The class of solenoidal planar-helical vector fields

The class of solenoidal vector fields whose lines lie in planes parallel to R ² is constructed by the method of mappings. This class exhausts the set of all smooth planarhelical solutions of Gromeka’s problem in some domain D ? R ³. In the case of domains D with cylindrical boundaries whose generators are orthogonal to R ², it is shown that the choice of a specific solution from the constructed class is reduced to the Dirichlet problem with respect to two functions that are harmonic conjugates in D ² = D∩R ²; i.e., Gromeka’s nonlinear problem is reduced to linear boundary value problems. As an example, a specific solution of the problem for an axisymmetric layer is presented. The solution is based on solving Dirichlet problems in the form of series uniformly convergent in \(\bar D^2\) in terms of wavelet systems that form bases of various spaces of functions harmonic in D ².     

On some solenoidal vector fields in spherical domains

We study the properties of some specific vector fields used in hydrodynamics and electrodynamics. We prove the possibility of mutual generation of two types of vector fields.     

Direct separation of two-dimensional vector fields into irrotational and solenoidal parts

An iterative technique is given for separating a two-dimensional vector into irrotational and solenoidal parts. In contrast to classical methods, the procedure operates directly on the orthogonal scalar components of the vector field, and gives the two separate fields as the result of a single sequence of operations. The manipulations are somewhat similar to a relaxation process. An example of the decomposition of a meteorological wind field is given.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

Poloidal and toroidal potentials for solenoidal fields

Summary Representation of solenoidal vector fields by poloidal and toroidal scalar fields relative to a fixed spatial direction is shown to be sufficient in the sense that any such vector field determines the scalar functions. The potentials are expressible by logarithmic potentials, and harmonic functions derived from them, over two-dimensional slices of the domain. The nature of the non-uniqueness of the scalar fields is discussed. Comments on applications, particularly in convection problems, are made.

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Zusammenfassung In vielen Problemen der Mechanik und Elektrodynamik ist die Darstellung divergenzfreier Vektorfelder in endlichen Gebieten durch poloidale und toroidale Potentiale ein wertvolles analytisches Hilfsmittel. In der Literatur findet man eine Vielzahl von speziellen Formen dieser Darstellung, die zur Behandlung spezieller Probleme geeignet sind. In der vorliegenden Arbeit wird gezeigt, daß jedes in einem GebietV und auf dessen Rand stetige und inV stetig differenzierbare und divergenzfreie Vektorfeldu durch solche Potentiale dargestellt werden kann. Diese Potentiale sind nicht eindeutig; die Art der Mehrdeutigkeit wird behandelt. Zum Schluß folgen einige Bemerkungen über Anwendungen.

     

Perturbation theorems for Hele-Shaw flows and their applications

In this work, we give a perturbation theorem for strong polynomial solutions to the zero surface tension Hele-Shaw equation driven by injection or suction, the so called Polubarinova–Galin equation. This theorem enables us to explore properties of solutions with initial functions close to polynomials. Applications of this theorem are given in the suction and injection cases. In the former case, we show that if the initial domain is close to a disk, most of the fluid will be sucked before the strong solution blows up. In the latter case, we obtain precise large-time rescaling behaviors for large data to Hele-Shaw flows in terms of invariant Richardson complex moments. This rescaling behavior result generalizes a recent result regarding large-time rescaling behavior for small data in terms of moments. As a byproduct of a theorem in this paper, a short proof of existence and uniqueness of strong solutions to the Polubarinova–Galin equation is given.     

Variant fountain theorems and their applications

In this paper we establish some variant fountain theorems without (P.S.)-type assumption. The abstract results will be used to study the symmetric nonlinear Schr?dinger equations and Dirichlet boundary value problems. Under no Ambrosetti–Rabinowitz's superquadraticity condition, we obtain infinitely many large energy and small negative energy solutions respectively. Received: 12 September 2000     

Minimal distortion morphs generated by time-dependent vector fields

A morph between two Riemannian n-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian n-manifolds and, with respect to these classes, prove the existence of minimal distortion morphs and diffeomorphisms. In particular, we consider the class of time-dependent vector fields (on an open subset Ω of Rn+1 in which the manifolds are embedded) that generate morphs between two manifolds M and N via an evolution equation, define the bending and the morphing distortion energies for these morphs, and prove the existence of minimizers of the corresponding functionals in the set of time-dependent vector fields that generate morphs between M and N and are L² functions from [0,1] to the Sobolev space .     

Some renewal theorems and their applications

Siberian Mathematical Journal -     

Generalized KKM theorems,minimax inequalities,and their applications

This paper extends the well-known KKM theorem and variational inequalities by relaxing the closedness of values of a correspondence and lower semicontinuity of a function. The approach adopted is based on Michael's continuous selection theorem. As applications, we provide theorems for the existence of maximum elements of a binary relation, a price equilibrium, and the complementarity problem. Thus our theorems, which do not require the openness of lower sections of the preference correspondences and the lower semicontinuity of the excess demand functions, generalize many of the existence theorems such as those in Sonnenschein (Ref. 1), Yannelis and Prabhakar (Ref. 2), and Border (Ref. 3).The author is grateful to Professor Franco Giannessi for helpful comments and suggestions.     

A similarity principle for complex vector fields and applications

This paper establishes a similarity principle for a class of non-elliptic, smooth complex vector fields in the plane. This principle is used to prove a uniqueness result for a nonlinear Cauchy problem.

     

The vector integral-equation method for computing three-dimensional magnetic fields

The vector integral-equation method for computing three-dimensional, quasistatic magnetic fields is developed with a view to its application to configurations of the type that occur in magnetic recording. Starting from appropriate Green-type vector integral relations for the magnetic-field quantities, the relevant integral equations are derived. Their numerical handling is briefly discussed.     

A free boundary problem for three-dimensional harmonic vector fields

We study in this paper a free boundary value problem ( FB ), where a region G_o in R ³ is determined by the condition that there exists a vector field v_o in G_o which satisfies div v_o = e_o, curl v_o = g_o in G_o and v_o = E on the boundary ?G_o with a given scalar function e_o and given vector fields g_o and E. We give two equivalent formulations for this problem. Then we characterize the solutions by a non-linear integral equation. In order to solve the latter by a Newton method we linearize this equation. We investigate the ensuing linear integral equation. In case of axisymmetric configurations this is a singular integral equation whose index can be easily determined from the given data. We obtain a related equation, if we try to construct a field v in a region G which is on the boundary perpendicular to a given field B . Finally we use this method to investigate an astrophysical problem, which arises in the theory of pulsar magnetospheres.     

Nonconvex separation theorems and some applications in vector optimization

Separation theorems for an arbitrary set and a not necessarily convex set in a linear topological space are proved and applied to vector optimization. Scalarization results for weakly efficient points and properly efficient points are deduced.