On Ω-closed sets and Ωs-closed sets in topological spaces

Summary New classes of sets called -closed sets and s-closed sets are introduced and studied. Also, we introduce and study -continuous functions and s-continuous functions and prove pasting lemma for these functions. Moreover, we introduce classes of topological spaces -T_1/2 and -T_s.     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

Projections on Lp-spaces of polyanalytic functions

The fundamental result: for an arbitrary bounded, simply connected domain in , the subspace L_n,m ^p() of the space L^p(, ) ( is the plane Lebesgue measure, p 1), consisting of the (m, n)-analytic functions in , is complemented in LP(, ) (a function f is said to be (m, n)-analytic if (^m+n/¯Z^mZⁿ)f=0 in ). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczyski, the space L_n,m ^P() is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic L^P-functions in is complemented in L^P(, ). This result has been known earlier only for smooth domains.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 15–33, 1991.     

An embedding theorem for a limiting exponent

We consider the function space B _p ^l () of functionsf(x), defined on the domain of a certain class and characterized by specific differential-difference properties in L_p(). We prove a theorem on the embedding B _p,q ^l () L_q in the case whenl=n/p –n/q >0 and its generalization for vectorl, p, q.Translated from Matematicheski Zametki, Vol. 6, No. 2, pp. 129–138, August, 1969.     

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. I

In the Banach space of functions analytic in a Jordan domain , we establish order estimates for the Kolmogorov widths of certain classes of functions that can be represented in by Cauchy-type integrals along the rectifiable curve = and can be analytically continued to or to .     

Partially ordered permutation groups

In this note we discuss permutation groups (G, ) in which the set admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of such that < implies g<g, for all and in andg inG. If the set admits aG-partial order which is a total order, then (G, ) is an O-permutation group (orderable permutation group).The main concern of this paper is the development of a foundation for partially ordered permutation groups analogous to the existing one for partially ordered groups, as found in Fuchs [2].     

Graphs of Constant Mean Curvature in Hyperbolic Space

We study the problem of finding constant mean curvature graphsover a domain of a totally geodesic hyperplane andan equidistant hypersurface Q of hyperbolic space. We findthe existence of graphs of constant mean curvature H overmean convex domains Q and with boundary for –H < H |h|, where H > 0 is the mean curvature of the boundary . Here h is the mean curvature respectively of the geodesic hyperplane (h= 0) and of the equidistant hypersurface (0 < |h|< 1). The lower bound on H is optimal.     

Classification analytique d'équations différentielles ydy +...=0 et espace de modules

Résumé Dans ce travail, on s'intéresse à la classification analytique de certains types d'équations différentielles de (²,0) de la forme =ydy+...=O Cette classification est en général donnée par celle de l'holonomie projective apparaissant dans la résolution de . Dans un cas spécial la classification est donnée par celles de l'holonomie associée à l'unique séparatrice de . On précise l'espace de modules et on prouve la rigidité générique de .

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In this paper we study the analytic classification of class of differential equations =ydy+...=O in (²,0). We prove that generically they are rigid. We also give the moduli spaces in special cases.

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L'auteur remercie l'IRMAR pour l'accueil qu'il a rencontré durant ses séjours (long et courts) à l'institut.     

Operatori lineariT-invarianti rispetto ad un gruppo di congruenze

Summary Let be an open subset of ⁿ, W_m() the linear space of m-vector valued functions defined on , G{} a group of orthogonal matrices mapping onto itself and T{T()} a linear representation of order m of G. A suitable groupC(G,T) of linear operators of W_m(), which leads to a general definition of T-invariant linear operator with respect to G, is here introduced. Characterization theorems concerning the linear differential and integral T-invariant operators are also given. When G is a finite group, projection operators are explicitly obtained; they define a «maximal» decomposition of W_m() into a direct sum of subspaces each of them invariant with respect to any T-invariant linear operator of W_m(). Some examples are givenc.

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Lavoro eseguito nell'ambito del progetto nazionale di ricerca «Analisi numerica e matematica computazionale» nell'anno 1985–86.     

Proper holomorphic mappings between some nonsmooth domains

Sunto In questo lavoro si studia il problema dell'existenza o meno di mappe olomorfe proprie, F: (p,q) (pq) dove (,) Cⁿ sono domini limitati, pseudoconvessi, non lisci, di Reinhardt con centro di simmetria sulla frontiera (per la definizione v. (2), introduzione). In caso affermativo si fornisee esplicitamente l'espressione di tali mappe (v, teoremi I e II nell'introduzione).

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The author, during the redaction of this paper, was supported by MPI (40% and 60%) funds.     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).     

Some overdetermined boundary value problems for harmonic functions

Summary In this paper we study various overdetermined problems involving harmonic functions. In particular, we show that if the second eigenfunctionu ₂ of the Stekloff eigenvalue problem in a bounded simply connected plane domain has a constant value of u ₂ on , then is a disk

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Résumé Cet article est consacré à l'étude de certains problèmes surdéterminés pour des fonctions harmoniques. En particulier, nous montrons que si le gradient de la seconde fonction propre du problème de Stekloff défini dans un domaine borné, simplement connexe du plan, a son module constant sur la frontière , alors est nécessairement un disque.

     

Compact Toeplitz operators via the Berezin transform on bounded symmetric domains

Let be an irreducible bounded symmetric domain of genusp, h(x, y) its Jordan triple determinant, andA ² () the standard weighted Bergman space of holomorphic functions on square-integrable with respect to the measureh(z, z) ^–p dz. Extending the recent result of Axler and Zheng for =D, =p=2 (the unweighted Bergman space on the unit disc), we show that ifS is a finite sum of finite products of Toeplitz operators onA ² () and is sufficiently large, thenS is compact if and only if the Berezin transform ofS tends to zero asz approaches . An analogous assertion for the Fock space is also obtained.The author's research was supported by GA AV R grant A1019701 and GA R grant 201/96/0411.     

Runge property and multiplicity of solutions for elliptic equations in ℝN with a monotone nonlinearity

In this paper, we describe a method for extending (in some approximated sense) solutions of a nonlinear P.D.E. on a domain , to solutions in a domain containing . Such an extension property, the Runge property, is well known for a large class of linear problems including elliptic equations. We prove here the Runge property for semilinear problems of the kind -u+g(u)=f, with f L _loc ¹ (^N). (As a consequence, we get infinitely many solutions for these problems). The proof is based on a homotopy method, and requires a refinement of the linear results: We prove that the Runge extension v on of a solution u in for a linear elliptic equation Lu=f can be choosen in order to depend continuously on u and the coefficients of L.     

Bounce problem with weak hypotheses of regularity

Summary In this paper the elastic bounce problem is formulated in very general hypotheses. More precisely we consider the motion of a material point constrained to move in a domain R ⁿ, bouncing against its boundary, and we suppose that is neither regular nor convex. Assuming that is in the class of p-convex sets introduced in [4] and C^0,1, an existence theorem is stated.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Convexes hyperboliques et fonctions quasisymétriques

Hyperbolic convex sets and quasisymmetric functions Every bounded convex open set of R ^m is endowed with its Hilbert metric d . We give a necessary and sufficient condition, called quasisymmetric convexity, for this metric space to be hyperbolic. As a corollary, when the boundary is real analytic, is always hyperbolic.In dimension 2, this condition is: in affine coordinates, the boundary is locally the graph of a C¹ strictly convex function whose derivative is quasisymmetric.      

Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let be a bounded domain in ⁿ (n3) having a smooth boundary, let be an essentially bounded real-valued function defined on × ^h, and let be a continuous real-valued function defined on a given subset Y of Y ^h. In this paper, the existence of strong solutions u W ^2,p (, ^h) W _o ^1,p (n/2<p<+) to the implicit elliptic equation (–u)=(x,u), with u=(u₁, u₂, ..., u_h) and u=(u ₁, u ₂, ..., u _h), is established. The abstract framework where the problem is placed is that of set-valued analysis.     

Functions with a finite dirichlet integral in a domain with cusp points on the boundary

Let be a domain in ⁿ, n >2, the boundary of which has a cusp point, pointing inside or outside the domain. The purpose of the paper is to characterize the traces on of the elements of the space H¹() of functions with a finite Dirichlet integral. As a consequence one establishes the existence of a linear continuous extension operator H¹ () H¹(ⁿ) under the presence of an interior cusp point on . Theorems on domains with cusps are proved with the aid of results on cylindrical domains. In the space of functions with a finite Dirichlet integral in the exterior or the interior of the cylinder one introduces the norm, depending on a small parameter and generating a norm of the trace on as an element of the quotient space. The latter is placed in correspondence with an explicitly described norm of functions on the boundary, uniformly equivalent relative to . One constructs an operator of extension of functions from the exterior of the cylinder to Rⁿ, preserving H¹, whose norm is uniformly bounded relative to . For the optimal operator of extension from the inside of the cylinder one finds the asymptotic behavior of the norm as 0. From these results there follow similar theorems on functions with a finite Dirichlet integral inside and outside a thin closed tube (of width ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 117–137, 1983.     

Certain problems of vector analysis

In the paper one gives an explicit method of constructing in the domain ^m m2 a vector field, having a prescribed divergencef in and taking prescribed values on the boundary . The differential properties of the field are faithfully determined by the smoothness off, and Simultaneously, one constructs the solutions of a series of other problems of vector analysis, which present an independent interest.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 65–85, 1984.