Showing papers on "Domain (mathematical analysis) published in 1985"

Weighted Sobolev Spaces

Abstract: Introduction Motivation Weight Domain Hardy Inequality Part One: POWER-TYPE WEIGHTS: Some Elementary Assertions Density of Smooth Functions Imbedding Theorems Miscellaneous Part Two: GENERAL WEIGHTS: Several Elementary Results Density of Smooth Functions Imbedding Theorems Part Three: APPLICATIONS: Formulation of the Problem Power-Type Weights General Weights References List of Imbeddings Index.

Spaces for the assessment of knowledge

Abstract: The information regarding a particular field of knowledge is conceptualized as a large, specified set of questions (or problems). The knowledge state of an individual with respect to that domain is formalized as the subset of all the questions that this individual is capable of solving. A particularly appealing postulate on the family of all possible knowledge states is that it is closed under arbitrary unions. A family of sets satisfying this condition is called a knowledge space. Generalizing a theorem of Birkhoff on partial orders, we show that knowledge spaces are in a one-to-one correspondence with AND/OR graphs of a particular kind. Two types of economical representations of knowledge spaces are analysed: bases, and Hasse systems, a concept generalizing that of a Hasse diagram of a partial order. The structures analysed here provide the foundation for later work on algorithmic procedures for the assessment of knowledge.

An existence result for nonlinear elliptic problems involving critical Sobolev exponent

Abstract: In this paper we consider the following problem: (1) { − Δ u − λ u = | u | 2 ⁎ − 2 ⋅ u u = 0 on ∂ Ω 2 ⁎ = 2 n / ( n − 2 ) where Ω ⊂ Rn is a bounded domain and λ ∈ R. We prove the existence of a nontrivial solution of (1) for any λ > 0, if n ⩾ 4.

Some entire functions with multiply-connected wandering domains

Abstract: A component U of the complement of the Julia set of an entire function ƒ is a wandering domain if the sets ƒn(U) are mutually disjoint, where n ∈ℕ and ƒn is the n-th iterate of ƒ. Examples are given of entire ƒ of order , which have multiply-connected wandering domains. An example is given where the connectivity is infinite.

Equations of Parabolic Type

Abstract: In this chapter we shall consider second-order parabolic equations and prove the unique solvability of the initial-boundary value problem in the domains Q T =, (x, t): x ∈ Ω,t ∈ (0, T) for the first, second, and third boundary conditions. We shall assume that the domain Ω is bounded, although all the results, except for the representation of solutions by Fourier series, will be valid for an arbitrary unbounded domain Ω. Moreover, the methods of solution given for bounded Ω are applicable to unbounded Ω (in particular, for Ω = R n ),but need minor modification which we shall point out.

Extension of proper holomorphic mappings past the boundary

Abstract: It is proved that proper holomorphic mappings between n-dimensional bounded pseudoconvex domains with real analytic boundaries extend holomorphically past the boundary whenever the target domain is strictly pseudoconvex.

Inequalities of bernstein type for derivatives of rational functions, and inverse theorems of rational approximation

Abstract: Let be the Hardy space of functions that are analytic in the disk and let be the derivative of of order in the sense of Weyl. It is shown, for example, that if is a rational function of degree with all its poles in the domain , then , where , , and depends only on and .Bibliography: 32 titles.

The regularity of the weighted Bergman projections

Abstract: In the paper the following fact is proved: If D is a smooth pseudoconvex bounded domain such that for some s > 0 there exists a compact operator Ts : W s (D)→Ws(D) solving the \(\bar \partial\)-problem \((\bar \partial T_s W = W)\), then for each \(w \in C^\infty (\bar D)\), the weighted Bergman projection with weight eW is a continuous operator from Ws(D) into Ws(D).

Method of conformal transformation for the finite‐element solution of axisymmetric exterior‐field problems

Abstract: The finite‐element method can be used for an approximate solution of axisymmetric exterior‐field problems by truncating the unbounded domain, or by applying various techniques of coupling a finite region of interest with the remaining far region, which is properly modelled. In this paper, we propose the solution of axisymmetric exterior‐field problems by using the standard finite‐element method in a bounded, transformed domain obtained by conformal mapping from the original, unbounded one. The transformed functionals have very simple expressions and the exact transforms of the original boundary conditions are used in the transformed domain. Consequently no approximation is introduced in the proposed method and improvements in the accuracy of the solution are obtained as compared with several other methods in common usage, especially with the truncated mesh technique. A few example problems are solved and the presented method is found to be simple and computationally highly efficient. It is particularly recommended for problems with material inhomogeneities and anisotropies within large regions.

Numerical Conformal Mapping of a Towel-Shaped Region onto a Rectangle

Abstract: If technical applications are involved, partial differential equations often have to be solved on a nearly rectangular domain (“towel”), one or more boundaries of which deviate from a straight line. A numerical method is presented for the mapping of such a region onto a rectangle. The calculation consists mainly in solving two coupled Laplace equations.

Sturmian theorems for second order systems

Abstract: Sturmian theorem are established for weakly coupled elliptic systems generated in a bounded domain by the expressions I1Ii = -xi; + Au, 12w -Aw + Bu, and Dirichlet boundary conditions. Here i\ denotes the Laplace operator, and A, B are nt X nt matrices. We do not assume that A, B are symmetric, but instead essentially require B irreducible and b,, 2, denote a bounded domain. We consider in H"12(G) the elliptic operators (1 ) 11u = -Af+ A-, (2) 12w -A-W + Bwi$, whereA = (aij), B = (bij) arem X m matrices, u= (ul,... ,uW)T, = .. are m vectors, A is the Laplace operator, and H1' 2(G) represents the usual Sobolev space with the norm IIUII1,2 = L IIUIi,||2 = f Du I .=1 jai1I We do not assume that A, B are symmetric. If m = 1 the typical Sturmian theorem states that every solution of 12w = 0 must vanish somewhere in G (or G) if A, B are suitably related and there exists a nontrivial solution u-> 0 of l1i lU 0 in Ho,2(G). Precise formulations of this and related results, together with the needed regularity hypotheses, may be found in the books by Swanson [12] and Kreith [9] and the more recent survey articles [13, 14]. Such results can, in particular, be used for eigenvalue comparison, and the above references contain several examples of such instances. It is the purpose of this paper to establish a Sturmian theorem and consequent eigenvalue comparison for the weakly coupled systems given by (1), (2). Unlike many earlier results, we do not deal with determinants of prepared matrix solutions or with h-oscillatory vector solutions (see the above references for clarification of these concepts). Instead, our results deal with the sign of the components of the solutions Received by the editors June 12, 1984 and, in revised form, August 14, 1984. 1980 Mathematics Subject Classification. Primary 35B05, 35P15, 35J55. Kei' words and phrases. Sturmian theorem, elliptic system, eigenvalue, positive operator. ,1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

Nontrivial solutions of elliptic boundary value problems with resonance at zero

Abstract: We consider a general nonlinear parabolic BVP (P) on a bounded and smooth domain Ω ⊂Rn, the nonlinearity being given by a functionf:\(\bar \Omega \times R \to R\). We impose various hypotheses on f: « nonresonance » (with respect to the linearized BVP) at infinity, « nonresonance » or «resonance» at zero. Using an extension of Conley's index theory to noncompact spaces, we prove the existence of equilibria of (P) (i.e. solutions of a corresponding elliptic equation), as well as trajectories joining some of these equilibria. The results obtained generalize earlier results of Amann and Zehnder (who were the first to apply the Conley index to elliptic equations), of Peitgen and Schmitt, and of this author.

Plane polygons and a conjecture of Blaschke’s

Abstract: The Radon transform of a plane domain is a random variable assigning to each line in the plane the chord length of its intersection with the domain. The probability distribution of this random variable does not characterize the domain, but it is shown to characterize a sufficiently asymmetric convex polygon. Under weaker assumptions, a convex polygon is characterized by this distribution, up to a finite number of rearrangements.

Existence of solutions in a cone for nonlinear alternative problems

Abstract: Using the alternative method we present sufficient conditions for the existence of positive solutions to nonlinear equations at resonance and extend a well-known result of Cesari and Kannan. Introduction. Cesari and Kannan [2] proved an abstract result in terms of the alternative method. Their result and some of its ramifications (see [1]) have been applied to a large class of problems at resonance to prove the existence of solutions. Let E be a Banach space. We say that C is a cone in E if C is a nonempty, convex subset of E such that XC c C for every X > 0. Here we prove the existence of solutions in a cone for equations at resonance of the form Lu = Nu, where L is a linear operator and N is a (nonlinear) operator. In the case when the cone is E, we obtain the well-known result of Cesari and Kannan [2]. In applications, for instance, if L is an elliptic operator on a bounded domain Q of R, one usually takes E as a subspace of L2(Q ) and the cone C = { u E E: u >, 0 a.e. in Q }. Also, our result is related to that of Gaines and Santanilla [3] concerning the existence of solutions in a convex set. Main result. Let E and F be Banach spaces with norms 11 IIE and 11 IIF' respectively. Let L: D(L) c E -* F be a linear operator and N: E -> F a continuous (nonlinear) operator such that N maps bounded sets into bounded sets. Assume that C is a cone in E and (1) there exists a continuous map -y: E -* C such that y(c) = c for every c E C, and -y maps bounded sets in E into bounded sets in E. In addition, suppose that L is a Fredholm map of index 0 and there exist projections P: E -* E, Q: F -F, and a linear map H: (I Q)F -(I P)E Received by the editors February 16, 1984 and, in revised form, June 17, 1984. 1980 Mathematics Subject Classification. Primary 47H15, 34B15, 34C15, 35G30, 35J40. ?01985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page

The bieberbach conjecture

Abstract: By the Riemann mapping theorem, every simply connected plane domain (except C itself) can be mapped conformally onto the uni t disk D. This powerful theorem often allows pure and applied mathematicians to reduce problems about plane domains to the special case of a disk or halfplane. To use this tool it is necessary to have information about the mapping function. The class S (for "schlicht") consists of all functions

Schiffer variation of complex structure and coordinates for Teichmüller spaces

Abstract: Schiffer variation of complex structure on a Riemann surfaceX 0 is achieved by punching out a parametric disc $$\bar D$$ fromX 0 and replacing it by another Jordan domain whose boundary curve is a holomorphic image of $$\partial \bar D$$ This change of structure depends on a complex parameter e which determines the holomorphic mapping function around $$\partial \bar D$$ It is very natural to look for conditions under which these e-parameters provide local coordinates for Teichmuller spaceT(X 0), (or reduced Teichmuller spaceT #(X0)) For compactX 0 this problem was first solved by Patt [8] using a complicated analysis of periods and Ahlfors' [2] τ-coordinates Using Gardiner's [6], [7] technique, (independently discovered by the present author), of interpreting Schiffer variation as a quasi conformal deformation of structure, we greatly simplify and generalize Patt's result Theorems 1 and 2 below take care of all the finitedimensional Teichmuller spaces In Theorem 3 we are able to analyse the situation for infinite dimensionalT(X 0) also Variational formulae for the dependence of classical moduli parameters on the e's follow painlessly

Global holomorphic approximations of CR functions on CR manifolds

Abstract: Under some geometric and analytic conditions on a generic CR submanifold , it is proved that there exists a neighborhood Ω of with the following property: For any strongly pseudoconvex domain any smooth CR function can be uniformly approximated on by functions holomorphic on Dbar:

Qualitative properties of the boundary derivative of the capacity potential for special classes of annular domains

Abstract: Let U(p) denote the capacity potential in an annular domain Ω (bounded by Jordan curves). We describe the qualitative behavior of ∥Δ∥ on the line segments of ∂Ω which are arbitrary, radial, or ν-directional (for some vector ν ≠ 0) in the respective cases where Ω is a convex, starlike, or ≠-convex annular domain. We apply these results to some free-boundary extremal problems in which the capacity plus weighted area functional is minimized in restricted classes of annular domains Ω.

Boundary conditions at inflow and outflow planes

Abstract: The problem of analyzing complex flows in infinite domains by replacing the infinite region by a bounded region is examined. The formulation of appropriate boundary conditions at inflow and outflow planes is discussed, and methods of verifying that the solution to the finite problem represents a valid approximation to the solution of the infinite problem are evaluated. A model problem for the creeping flow of a Newtonian fluid is solved, and it is shown that utilization of a finite domain leads to some inaccuracies in the evaluation of pressure gradients and pressure drops. It is further shown for the model problem that it is possible to obtain seemingly acceptable solutions for the finite flow field which are, in reality, poor approximations to the solution of the infinite problem.

An Efficient Numerical Method for Determination of Shapes, Sizes and Orientations of Flaws for Nondestructive Evaluation

Abstract: The generalized pulse-spectrum technique (GPST)1 is a versatile and effcient iterative numerical algorithm for solving inverse problems (to determine the unknown coefficients, initial-boundary values, sources, and geometries of the space domain from the additionally measured data in the space-time domain or the space-complex frequency domain) of a system of nonlinear partial differential equations. Mathematically, inverse problems of partial differential equations can be formulated as ill-posed nonlinear operator equations. It is important to point out that the GPST is not a single narrowly defined iterative numerical algorithm but a broad class of iterative numerical algorithms based on the concept that either the nonlinear operator equation is first linearized by any one of the Newton-like iteration methods and then each iterate is solved by using a stabilizing method, e.g., the Tikhonov’s regularization method2, or the stabilizing method is first applied to the nonlinear operator equation and then the regularized nonlinear problem is solved by using a Newton-like iteration method. Hence different choices of various Newton-like iteration methods and stabilizing methods lead to different special forms of GPST, and the efficiency of GPST will then depend upon the particular choices of them and how efficiently one can treat every minute step in the numerical algorithm.

Solvability of a first order system in three-dimensional non-smooth domains

Abstract: A system of first order partial differential equations is studied which is defined by the divergence and rotation operators in a bounded nonsmooth domain $\Omega\subset \bold R^3$. On the boundary $\delta\Omega$, the vanishing normal component is prescribed. A variational formulation is given and its solvability is investigated.

A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation

Abstract: : Several preconditioned conjugate gradient (PCG)-based domain decomposition techniques for self-adjoint elliptic partial differential equations in two dimensions are compared against each other and against conventional PCG iterative techniques in serial and parallel contexts. The authors consider preconditioners that make use of fast Poisson solvers on the subdomain interiors. Several preconditioners for the interfacial equations are tested on a set of model problems involving two or four subdomains, which are prototype of the stripwise and boxwise decompositions of a two-dimensional region. Selected methods have been implemented on the Intel Hypercube by assigning one processor to each subdomain, making use of up to 64 processors. The choice of a 'best' method for a given problem depends in general upon: (a) the domain geometry, (b) the variability of the operator, and (c) machine characteristics such as the number of processors available and their interconnection scheme, the memory available per processor, and communication and computation rates. Emphasized is the importance of the third category, which has not been as extensively explored as the first two in the domain decomposition literature to date. (Author)