Showing papers on "Domain (mathematical analysis) published in 1971"

Nonlinear Functional Analysis and Nonlinear Integral Equations of Hammerstein and Urysohn Type

Abstract: Publisher Summary This chapter discusses nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type. The theory of nonlinear integral equations of Hammerstein type has been, since its inception in the paper of Hammerstein, one of the most important domains of application of the ideas and techniques of nonlinear functional analysis, second only to the theory of solutions of boundary value problems for nonlinear partial differential equations. The development of the fixed point and degree theory for compact nonlinear mappings in Banach spaces was strongly influenced, in its form, by the theory of nonlinear integral equations and was directly applied to this domain and many others. The chapter presents a unified development of the theory of the Hammerstein equation using the theory of the topological degree for mappings of the form I - C with C compact as well as the basic theory of monotone nonlinear mappings from X to X*.

On the mean-value property of harmonic functions

Abstract: In this note we show that if the areal mean-value theorem holds for a plane domain (subject to a mild regularity con- dition) for all integrable harmonic functions, then the domain must be a disk. It is also shown that if a plane domain with finite area has at least two boundary components which are continua then the mean-value property cannot hold for the class of all integrable har- monic functions with single-valued harmonic conjugates. 1. In 1962 Epstein1 (2) proved the following theorem: "Let D be a simply connected domain of finite area and t a point of D such that, for every function u harmonic in D and integrable over D, the mean- value of u over the area of D equals u(t). Then D is a disk and t its center." In a later paper Epstein and Schiffer extended the above result to domains in Euclidean space En replacing the simple connec- tivity hypothesis of the earlier paper by the assumption that the complement of D possess a nonempty interior. Nevertheless, the following theorem strongly suggests that, for plane regions at least, the simple connectedness of D is a necessary condition for the mean- value property to hold. Theorem 1. Let D be a plane domain of finite area having at least two boundary components 71 and 72 which are continua. Denote by 3C the class of functions u harmonic in D and integrable over D; and ey IFC3C the subclass consisting of functions possessing single-valued harmonic conjugates in D. Then i (and a fortiori 3C) does not satisfy the mean- value property at any point tED; that is, it is not the case that there exists a tED such that, for all uE$,

An iterative method of solving elliptic net problems

Abstract: The application of the iterative process proposed by R. P. Fedorenko to the case of finite-difference equations for an arbitrary elliptic equation in a rectangle was justified in [1]. In this paper the results of [1] are transferred to the third boundary value problem in an arbitrary smooth domain.

On a forward-backward parabolic equation

Abstract: Boundary value problems for the equation $$\operatorname{sgn} (x) \cdot u_y - u_{xx} + ku = f(x,y)$$ (where k is a positive constant and † is a given function) are investigated. The domain of the solutions will be the whole upper half-plane y>0, or the half-plane y>0 cut along the positive y-axis. We are interested in square integrable solutions u, with square integrable generalized derivatives uy and uxx. Existence theorems are proved, with an integral equations technique. Thus a theory is developed of Wiener-Hopf integral equations of the first kind with solutions belonging to Sobolev spaces.

EXACT HÖLDER ESTIMATES FOR THE SOLUTIONS OF THE \bar\partial-EQUATION

Abstract: The -equation with bounded right side in a strictly pseudoconvex domain has a solution which satisfies the Holder condition with exponent 1/2.

Problemi al contorno con condizioni omogenee per le equazioni quasi-ellittiche

Abstract: We are concerned with non-variational boundary value problems, with omogeneus boundary conditions, for linear partial differential equations of quasi-elliptic type in a bounded domain Θ in Rn. It is well known that some of difficulties which arise in treating such problems, in comparison with « regular » elliptic problems, are connected with the presence of angular points in Θ: let us point out withB. Pini [32] that « a bounded domain for which it is possible to assign a correct boundary value problem for a quasi-elliptic but not elliptic equation always has angular points ». We suppose Θ is a cartesian product of a finite number of open sets and, in order to overcome the difficulties attached to the presence of angular points in Θ, taking as a model the two previous papers[33], [34] devoted to elliptic problems with singular data, we investigate the problem within suitable Sobolev weight spaces, connected with the angular points of Θ and included in the ones we have studied in[35]. Within such spaces we get existence and uniqueness theorems.

Dynamic programming and the solution of the biharmonic equation

Abstract: The numerical solution of the biharmonic equation in a rectangular domain is presented in the context of continuous dynamic programming techniques. The equations are specialized to the solution of elastic rectangular plates. A suitable approximate expression of a certain functional equation containing derivatives only in one direction is used to derive equations for the stiffness and flexibility matrices of the plate. It is shown that those matrices satisfy matrix Riccati equations subject to suitable initial conditions. It is also shown that the condition of optimality in the Hamilton-Jacobi-Bellman equation directly expresses a classical variational principle, i.e. the principle of complementary energy. Some numerical examples are finally presented.

On local smoothness of generalized and hypoellipticity of second order differential equations

Abstract: The paper gives a survey of results on hypoellipticity of second order differential operators. In particular, our new theorems together with earlier results give a necessary and sufficient condition for hypoellipticity of a general second order differential operator with analytic coefficients, provided that it is not fully degenerate at any point of the domain. We dedicate this paper to I. G. Petrovskii on the occasion of his seventieth birthday.

Green's relations in finite function semigroups

Abstract: Letℱn be the set of all partial functions on ann-element setXn, i.e., the set of all functions whose domain and range are subsets ofXn. Green's equivalence relationsℛ, ℒ, ℋ andℋ are considered, and the number and cardinality of the corresponding equivalence classes are determined. The number of idempotent and generalized idempotent elements inℱn is also determined.

An interior a priori estimate for parabolic difference operators and an application

Abstract: A general class of finite-difference approximations to a parabolic system of differential equations in a bounded domain S is considered. It is shown that if a solution Uh of the discrete problem converges in a discrete L2 norm to a solution U of the con- tinuous problem as the mesh size h tends to zero, then the difference quotients of Uh con- verge to the corresponding derivatives of U, the convergence being uniform on any com- pact subset of S. In particular, Uh converges uniformly on compact subsets to U as h tends to zero, provided there is convergence in the discrete L2 norm. The main part of the paper is devoted to the establishment of an a priori estimate for the solutions of the dis- crete problem. This estimate is then used to derive the stated result.

The space-time domain magnetic vector potential integral equations

Abstract: The space-frequency domain magnetic vector potential integral equations for the current distribution on wire antennas (Hallen's integral equation and Mei's generalization) are used to derive their counterparts in the space-time domain. Equations for both straight and curved wires are readily derived using a Fourier transform.

On quasi-analytic vectors for dissipative operators

Abstract: In this note we shall prove that a closed dissipative operator A with dense domain in a hilbert space H generates a contraction semigroup if the set A kx; k = 0, 1, 2, , x is quasi-analytic}

Conditions for continuity of certain open monotone functions

Abstract: In this paper continuity of certain open monotone functions is obtained by assuming for the domain and/or range various combinations of the properties of a metric continuum, regular metric continuum, semilocal connectedness, and hereditary local connectedness. An open monotone connected function from a hereditarily locally connected separable metric continuum onto a separable metric continuum is continuous. If the domain is a regular separable metric continuum, an upper semicontinuous decomposition and resulting monotone-light factorization yield continuity of an open monotone function with closed point inverses. By a continuum is meant a compact connected space. A function f is monotone if point inverses are connected. If f is a function from X onto Y, the component decomposition X' of X induced byf is the collection of all components of sets of the form f 1(y), where y varies over Y. A function is connected if it takes connected sets onto connected sets. A continuum is regular provided every point has arbitrarily small open neighborhoods with finite boundaries [4]. It should be noted that this is not the same as a regular topological space as usually defined. THEOREM 1. If f is an open monotone connected function from the 1st countable space X onto the 1st countable semilocally connected space Y, then f is continuous. PROOF. If f is not continuous there exists an open set U in Y such thatf-'(U) is not open in X. Hence there is a point xCf1(U) and a sequence {xn } of distinct points in X -f'(U) such that xn--+x. Since Y is semilocally connected there exists an open set VC U such that YV has only a finite number of components. Since f(xn)E V for all n it follows that some component C of YV contains f(xn) for infinitely many n. By Theorem 2 of [I], ft1(C) is connected, and x is a limit point of f-1(C). Hence ft-(C)U{x} is connected but Received by the editors October 19, 1970. AMS 1969 subject classifications. Primary 5460; Secondary 5455.

Bergman minimal domains in several complex variables

Abstract: K. T. Hahn has obtained the inequality between the Jacobians of a biholomorphic mapping and a holomorphic automorphism of a Bergman minimal domain. This paper extends Hahn's result. Some inequalities concerning Jacobians of the mappings of minimal domains onto another minimal domain are considered, and an example is given.

Remark on Siegel domains of type

Abstract: Bounded symmetric domains have standard realizations as "Siegel domains of type III." Pjateckil-Sapiro has introduced a more restrictive notion of "Siegel domain of type III." Here we give a direct proof that those standard realizations satisfy the additional conditions of the new definition.

Bounded holomorphic functions of several complex variables. I

Abstract: A domain of bounded holomorphy in a complex analytic manifold is a maximal domain for which every bounded holomorphic function has a bounded analytic continuation. In this paper, we show that this is a local property: if, for each boundary point of a domain, there exists a bounded holomorphic function which cannot be continued to any neighborhood of the point, then there exists a single bounded holomorphic function which cannot be continued to any neighborhood of the boundary points. Introduction.. Let X be a topological space. A subset D of X is said to be a region if it is open and i. is said to be a domain if it is open and connected. We denote by N(p) a fundamental system of open neighborhoods of p, where p E X. 1. DEFINITION. Let X be a topological space and U be an open subset of X. Let C(U) be the family of all continuous complex-valued functions on U, then C(U) is an algebra with 1, and it is equipped with the topology of uniform convergence on compact subsets of U. For a pair of open subsets U and V in X such that Vc U we define 7ru,: C(U) -C(V) by Tuvf=f I V. Let A(U) be a subalgebra of C(U) with I and we assume that ITUVA(U)cA(V); then we call A={A(U), 7Tuv} a presheaf of algebras of functions. A presheaf A has the local belonging property if, for all open sets U of X andf in C(U), for every p E U there is V E N(p), Vc U, such thatfI Vc A(v); thenfc A(U). A sheaf A of algebras of functions is a presheaf of algebras of functions with the local belonging property. A is said to be a ringed structure on X and the pair (X, A) is said to be a ringed space. The functions in A(U) are A-holomorphic functions. We note that A(U) has the relative topology induced by the topology on C(U). A ringed structure A on X is an n-dimensional complex analytic structure on X if for all x E X there are U E N(x) andf1,.. .f,f E A(U) such that F=(fi, ... .,n): U--* Cn Received by the editors February 5, 1970 and, in revised form, September 16, 1970. AMS 1970 subject classifications. Primary 32D15; Secondary 32D05.

Difference schemes on a self-organizing set of calculated points in two-dimensional simply connected domains of arbitrary form☆

Abstract: The numerical solution of some two-dimensional problems for partial differential equations by the usual difference schemes on rectangular nets suffers from a number of difficulties. For example, the use of rectangular nets for domains with curvilinear boundaries destroys the standardisation of the calculation and raises the question of the difference approximation of the equations close to the boundary. The obtaining of solutions with an essentially different kind of variation in the calculated domain is usually also hindered. Such difficult are of a fundamental nature and prevent the solution of a number of problems. In this connection at the present time it was felt to be necessary to search for new ways of constructing numerical methods free from the deficiencies indicated. Among the investigations in this direction we may mention papers [ldl.

Stability of feedback systems with pulse-frequency modulation†

Abstract: In the past few years analysis of feedback systems with pulse-frequency modulation has been investigated using the continuous-time domain approach. This paper develops a non-linear discrete equivalence of a class of pulse-frequency modulation feedback systems. Using the non-linear discrete equivalence, stability of feedback systems with pulse-frequency modulation is analysed through the second method of Lyapunov.

The domain rank of open surfaces of infinite genus

Abstract: In a recent paper it was shown that an open surface, i.e. a connected 2-manifold without boundary, has finite domain rank if and only if it has finite genus. In the present paper, it is shown that the domain rank of any open surface of infinite genus is countably infinite.