A Lagrange multiplier theorem and a sandwich theorem for convex relations.
An open mapping theorem for analytic multifunctions
The paper gives sufficient conditions for projections of certain pseudoconcave sets to be open. More specifically, it is shown that the range of an analytic set-valued function whose values are simply connected planar continua is open, provided there does not exist a point which belongs to boundaries of all the fibers. The main tool is a theorem on existence of analytic discs in certain polynomially convex hulls, obtained earlier by the author.
Continuous dependence for solution classes of Euler-Lagrange equations generated by linear growth energies
In this paper, a one-dimensional Euler-Lagrange equation associated with the total variation energy, and Euler-Lagrange equations generated by approximating total variations with linear growth, are considered. Each of the problems presented can be regarded as a governing equation for steady-states in solid-liquid phase transitions. On the basis of precise structural analysis for the solutions, the continuous dependence between the solution classes of approximating problems and that of the limiting...
Correspondence between subadditive integrals and small systems
Discontinuous wave equations and a topological degree for some classes of multi-valued mappings
The Leray-Schauder degree is extended to certain multi-valued mappings on separable Hilbert spaces with applications to the existence of weak periodic solutions of discontinuous semilinear wave equations with fixed ends.
Existence and density results for retarded subdifferential evolution inclusions
In this paper we study Cauchy problems for retarded evolution inclusions, where the Fréchet subdifferential of a function F:Ω→R∪{+∞} (Ω is an open subset of a real separable Hilbert space) having a φ-monotone subdifferential of oder two is present. First we establish the existence of extremal trajectories and we show that the set of these trajectories is dense in the solution set of the original convex problem for the norm topology of the Banach space C([-r, T₀], Ω) ("strong relaxation theorem")....
Fredholm alternatives and surjectivity results for multivalued -proper and condensing mappings with applications to nonlinear integral and differential equations
Global φ-attractor for a modified 3D Bénard system on channel-like domains
In this paper we prove the existence of a global φ-attractor in the weak topology of the natural phase space for the family of multi-valued processes generated by solutions of a nonautonomous modified 3D Bénard system in unbounded domains for which Poincaré inequality takes place.
Graphs of finite mass which annot be approximated in area by smooth graphs.
Harmonic morphisms onto Riemann surfaces and generalized analytic functions
We study harmonic morphisms from domains in and to a Riemann surface , obtaining the classification of such in terms of holomorphic mappings from a covering space of into certain Grassmannians. We show that the only non-constant submersive harmonic morphism defined on the whole of to a Riemann surface is essentially the Hopf map.Comparison is made with the theory of analytic functions. In particular we consider multiple-valued harmonic morphisms defined on domains in and show how a cutting...
On a two-point boundary value problem for second-order differential inclusions on Riemannian manifolds.
On differential equations and inclusions with mean derivatives on a compact manifold
We introduce and investigate a new sort of stochastic differential inclusions on manifolds, given in terms of mean derivatives of a stochastic process, introduced by Nelson for the needs of the so called stochastic mechanics. This class of stochastic inclusions is ideologically the closest one to ordinary differential inclusions. For inclusions with forward mean derivatives on manifolds we prove some results on the existence of solutions.
On differential inclusions of velocity hodograph type with Carathéodory conditions on Riemannian manifolds
We investigate velocity hodograph inclusions for the case of right-hand sides satisfying upper Carathéodory conditions. As an application we obtain an existence theorem for a boundary value problem for second-order differential inclusions on complete Riemannian manifolds with Carathéodory right-hand sides.
On the differentiability of multivalued mappings. II.
On the embedding theorem
Spaces of upper semicontinuous multi-valued functions on complete metric spaces
Let X = (X,d) be a metric space and let the product space X × ℝ be endowed with the metric ϱ ((x,t),(x’,t’)) = maxd(x,x’), |t - t’|. We denote by the space of bounded upper semicontinuous multi-valued functions φ : X → ℝ such that each φ(x) is a closed interval. We identify with its graph which is a closed subset of X × ℝ. The space admits the Hausdorff metric induced by ϱ. It is proved that if X = (X,d) is uniformly locally connected, non-compact and complete, then is homeomorphic to a...
Stochastic differential inclusions of Langevin type on Riemannian manifolds
We introduce and investigate a set-valued analogue of classical Langevin equation on a Riemannian manifold that may arise as a description of some physical processes (e.g., the motion of the physical Brownian particle) on non-linear configuration space under discontinuous forces or forces with control. Several existence theorems are proved.
Sur l'évaluation de certains volumes
The Demyanov metric and some other metrics in the family of convex sets
We describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.
Undirected and directed graphs with near polynomial growth
The growth function of a graph with respect to a vertex is near polynomial if there exists a polynomial bounding it above for infinitely many positive integers. In the paper vertex-symmetric undirected graphs and vertex-symmetric directed graphs with coinciding in- and out-degrees are described in the case their growth functions are near polynomial.