Showing papers on "Domain (mathematical analysis) published in 1976"

Data Types as Lattices

Abstract: The meaning of many kinds of expressions in programming languages can be taken as elements of certain spaces of “partial” objects. In this report these spaces are modeled in one universal domain ${...

On the angle condition in the finite element method

Abstract: The finite element procedure consists in finding an approximate solution in the form of piecewise linear functions, piecewise quadratic, etc. For two-dimensional problems, one of the most frequently used approaches is to triangulate the domain and find the approximate solution which is linear, quadratic, etc., in every triangle. A condition which is considered essential is that the angle of every triangle, independent of its size, should not be small. In this paper it is shown that the minimum angle condition is not essential. What is essential is the fact that no angle is too close to $180^ \circ $.

Statistical mechanics of two‐dimensional vortices in a bounded container

Abstract: The equilibrium statistics of a large number of two‐dimensional point vortices of arbitrary sign, evolving in an arbitrary domain closed by a bounded curve, are investigated in the microcanonical formulation. The resulting differential equations for the spatial distribution function of the vortices are numerically integrated in various cases and the associated thermodynamic functions are computed. The case of a globally neutral, spatially uniform distribution is particularly studied for its connections with two‐dimensional turbulence and the use of the random phase approximation. Some numerical simulations of vortex motion in a circular domain support the theoretical development.

A Time-Dependent Lateral Boundary Scheme for Limited-Area Primitive Equation Models

Abstract: Before high-resolution numerical models can be of use operationally, they must be restricted to a limited domain, thus necessitating lateral boundary conditions which allow the changes outside the limited domain to influence the results while not contaminating the forecast with spurious boundary-reflected energy. Such a set of time-dependent lateral boundary conditions are presented in this paper. This boundary condition set is investigated using the linear analytic and finite-difference advection equations, the non-linear finite-difference shallow-water equations, and the hydrostatic primitive equations. The results illustrate how the boundary condition transforms long- and medium-length interior advective and gravity waves into short waves which can then be removed by a low pass filter, thereby giving the appearance that the exiting wave simply passed through the boundary. The results also indicate that large-scale advective and gravity waves enter the forecast domain with little degradation. T...

Nonlinear Elliptic Equations with Nonlinear Boundary Conditions

Abstract: Publisher Summary This chapter focuses on nonlinear elliptic equations with nonlinear boundary conditions, and discusses mildly nonlinear elliptic boundary value problems (BVPs). For the study of the stability of the solutions of the parabolic initial-boundary value problem, one has to have a good knowledge of the steady states, that is, of the solutions of the elliptic BVP. The most interesting case occurs if the elliptic BVP has several distinct solutions. Some of the tools, namely the theory of increasing, completely continuous maps in ordered Banach spaces are then used to enlarge the domain of applicability of the general existence theorem by deriving simple sufficient criteria for the existence of sub- and supersolutions. In addition, a nonexistence and a general uniqueness theorem are derived. Also, to demonstrate the power of this abstract approach, a multiplicity result, namely a criterion guaranteeing the existence of atleast three distinct solutions, is derived. The chapter also discusses the main results for the nonlinear BVP, a fundamental a priori estimate for the solutions of the linear BVP, the equivalent fixed point equation, and multiplicity theorem.

6.—Oscillation Theory for Semilinear Schrödinger Equations and Inequalities

Abstract: Sufficient conditions are derived for every solution of a nonlinear Schrodinger equation (or inequality) to be oscillatory in an exterior domain of En. Such results apply in particular to the n-dimensional Emden-Fowler equation. The method involves oscillatory behaviour of solutions of a nonlinear ordinary differential inequality satisfied by the spherical mean of a positive solution of the Schrodinger equation.

On computing the range of a rational function of n variables over a bounded region

Abstract: We seek an efficient method for computing the range of values of a function ofn variables over a bounded domain.

A boundary value problem for the elliptic equation of second order in a domain with a narrow slit. 1. the two-dimensional case

Abstract: An elliptic equation is considered in a domain that is obtained from a two-dimensional domain by removing a neighborhood of a segment. It is assumed that this neighborhood contracts to the segment as . An asymptotic expansion of the solution of the first boundary value problem in is constructed and justified. Bibliography: 5 titles.

Uniform And Tangential Approximations By Meromorphic Functions on Closed Sets

Abstract: Let G be an (open) domain in the finite complex plane and F a relatively closed proper subset of G. We denote by M(G) the set of functions meromorphic on G and as usual by R(K) (for a compact set K) the set of uniform limits of rational functions without poles on K.

Pseudoconvex Domains on a Kähler Manifold with Positive Holomorphic Bisectional Curvature

Abstract: Let M be an ^-dimensional complex manifold with a real analytic kahler metric. Throughout this paper, a kahler manifold is assumed to have a real analytic kahler metric without mentioning it. A relatively compact domain D in M is called a pseudoconvex (resp. strongly pseudoconvex) domain if there exist a neighborhood U of p and a pseudoconvex (resp. strongly pseudoconvex) function (p on U satisfying Df] U= {(p<$\ for each boundary point p^dD. We write simply s-pseudoconvex domains (resp. functions) for strongly pseudoconvex domains (resp. functions). Note that pseudoconvex domains are not always Stein manifolds. The purpose of the present paper is to show the following theorem:

A formula and estimates for the solutions of the tangential cauchy–riemann equation

Abstract: In this paper an explicit integral formula is derived for solutions of the homogeneous tangential Cauchy–Riemann equation on the boundary of a strongly pseudoconvex domain in , and best possible estimates are obtained. Using this formula, some questions on the approximation of holomorphic functions on the boundary of a strongly pseudoconvex domain are studied. Bibliography: 27 titles.

Light open and open mappings on manifolds. II

Abstract: Sufficient conditions are given for the existence of light open mappings between p.l. manifolds. In addition, it is shown that a mapping f from a pl. manifold Mm, m a 3, to a polyhedron Q is homotopic to an open mapping of M onto Q iff the index of f.Qr1(rM)) in irl(( is finite. Finally, it is shown that an open mapping of Mm onto a p.l. manifold Nn, n > m > 3, can be approximated by a light open mapping of M onto N. In [19], D. Wilson constructs examples of light open mappings (with each point inverse a Cantor set) from any 3 manifold onto any n cell (n > 3) and he constructs examples of monotone open mappings of any p1. manifold Mm (m > 3) onto any n cell (these results answered questions raised by Eilenberg in [3]). In the first paper in this series [24], the author gave a complete analysis of the existence of monotone and monotone open mappings from manifolds onto polyhedra. In this paper, we give a complete analysis of the existence of open mappings from manifolds onto polyhedra (using results from [24] and from the theory of covering spaces); however, the principal content of this paper is the technique developed in ?5 for constructing light open mappings between manifolds (with each point inverse a Cantor set). The techniques used in this paper are inspired by those of D. Wilson in [19] and [18 ]; indeed, the many similarities are apparent. The "key" result which enables us to remove the assumption (of Wilson in [19]) that the domain manifold have dimension three is contained in the appendix (it is necessary to study ?5 in order to understand the relevance of the appendix). The philosophy behind removing the assumption that the image is a cell is exactly the same as in [24] (however, we must assume the image is a manifold). In addition, the technical difficulties encountered in ?5 are numer-

Regularity of solutions of the Stokes problem in a polygonal domain

Abstract: Publisher Summary This chapter discusses several regularity results for the Stokes problem in a polygonal domain. Analogous regularity results for solutions of a single 2mth order elliptic equation in a polygonal domain have been extensively developed. Regularity results are of fundamental importance in the analysis of numerical methods for the Stokes problem. Also, regularity results are used in analyzing the stability of stationary solutions of the Navier-Stokes equations. The results presented in the chapter are all based on the method used by Kondratev to study the regularity of a single 2mth order elliptic equation. The results depend in an essential way on the spectral properties of a system of ordinary differential equations, which is associated with the Stokes equations; the chapter discusses this dependence. The chapter also discusses two low order regularity results and two higher order results.

A strange bounded smooth domain of holomorphy

Abstract: Introduction. Let 12 C C" be a bounded pseudoconvex domain. Does U have a neighborhood basis consisting of pseudoconvex domains? It is well known that the answer to this question is, in general, "no". But it has been an open problem, at least since 1933 when the fundamental paper [1] of H. Behnke and P. Thullen appeared, whether the answer might be in the affirmative under the additional hypothesis of smoothness of the boundary b£2. The main purpose of this note is to give an example of a bounded pseudoconvex domain £lx C C 2 with smooth boundary that nevertheless does not have a Stein neighborhood basis. Additional hypotheses that guarantee the existence of such a basis are given in [3]. The constructed domain £2X has some more strange properties. In particular, the conjecture of R. 0 . Wells [5, Conjecture 3.1], does not hold for £lx (Theorem 2) and the boundary bQ,1 cannot be described by a smooth function with positive semidefinite Leviform on the whole C at each point p E b£2 r Together with the beautiful example of Kohn and Nirenberg [4] , this domain Q,j shows that bounded smooth domains in C can have quite different analytic properties than strictly pseudoconvex domains. The proofs of the theorems announced in this note will be contained in a later paper of the authors. Definition of £2X. We fix a smooth function X:R —• R + U {0} with the properties X(x) = 0 for x 0 for x > 0, and such that X is "sufficiently" convex. For r > 1 we define a family of smooth functions p : ( C { 0 } ) x C —• R by putting

On generalized resolvents

Abstract: Let T be a bounded linear operator on a Hilbert space and p,.(T) the Fredholm domain of T. It is shown that a generalized resolvent can be constructed for T in PF(T) which verifies the resolvent identity except for an at most countable set of points which are close to the boundary of PF( T). Let T be a bounded linear operator on a Hilbert space H. In case the range of T is a closed subspace of H, then an operator F will be called a generalized inverse of T when FT is a projection onto the orthogonal complement of the kernel of T and TF is a projection onto the range of T. Unless T is invertible, then a generalized inverse is not unique. Let Q be a domain in the complex plane C such that for every X in X, the operator X T has closed range. An operator valued function F defined on 9 is called a generalized inverse function for T in 9 in case, for every X in X, F (X) is a generalized inverse of X T. A generalized inverse function F for T on an open set 6 is said to verify the resolvent identity on X, when for every pair X, ft in a component of (1) F(A) F( ) = F X)F(X)F( K)* A continuous generalized inverse function, for an operator T on an open set X, which verifies the resolvent identity on 6 will be called a generalized resolvent on . This note is concerned with the construction of generalized resolvents on open subsets of the Fredholm domain of a bounded operator T. Recall that an operator T is called semi-Fredholm in case T has closed range and the dimension of at least one of ker(T) or ker(T*) is finite; here, ker denotes kernel and T* is the adjoint of T. If T has closed range and both ker(T) and ker(T*) are finite dimensional, then T is called a Fredholm operator. The semi-Fredholm domain of T is the set PS-F(T) = {A E C: X T is semi-Fredholm} and the Fredholm domain of T is the set PF(T) = {x E C: X T is Fredholm}. There is one obvious obstruction to constructing a generalized resolvent for T on all of PS_F(T). In PS_F(T) there is an at most countable set where the function m(X) = minimum dimension[ker(Q T), ker(K T)*3 is discontinuous [3, Proposition 2.6], [5], [6]. This set will be denoted by PF(T) and is referred to as the set of singular points in the semi-Fredholm Received by the editors December 13, 1974 and, in revised form, July 21, 1975. AMS (MOS) subject classifications (1970). Primary 47A10; Secondary 47A25. '? American Mathematical Society 1976

Isoperimetric inequalities for a class of nonlinear parabolic equations

Abstract: Existence theorems and a priori bounds for a class of nonlinear parabolic equations are established. By means of an iteration process and symmetrization methods the solution in an arbitrary domain is compared with the one for the sphere of the same volume. It is shown that among all domains of given volume the sphere is the least stable.

Difference approximation methods for problems of mathematical physics

Abstract: In the paper we construct and investigate conservative schemes for elliptic equations in an arbitrary domain. To obtain difference approximations in the case of equations with mixed derivatives and with boundary conditions of the third kind, the concept of a vector scheme proves to be useful. Vector difference schemes are constructed by means of the integro-interpolation method (balance method). To obtain economical algorithms for the solution of many-dimensional parabolic problems we use the method of summary approximations, which leads to additive schemes and vector additive schemes. In particular, we construct economical additive vector schemes for parabolic equations with boundary conditions of the third kind in an arbitrary domain.

Solution of Hallam’s problem on the terminal comparison principle for ordinary differential inequalities

Abstract: We solve affirmatively the open problem raised by Hallam [31 and we apply this result to classical differential inequalities as well as to get existence and uniqueness theorems for the terminal value problem for ordinary differential equations. Hallam in [3] points out that a comparison principle on terminal values for ordinary differential inequalities stated by Mamedov [8] has wrong proof and, in establishing a weaker form of it, raises the problem of proving Mamedov's theorem in full generality. The aim of this paper is to answer affirmatively this question. The comparison theorem proved here is exactly the analogue for terminal values of the classical comparison theorem for initial values and therefore it is even more general than Mamedov's statement. This result is applied to improve some well-known propositions on differential inequalities for initial values as well as to prove some uniqueness and existence theorems for the terminal value problem for ordinary differential equations. In particular, an analogue of the Peano-Kneser theorem is proved: the set of solutions of terminal value problems is connected under conditions like those of Brauer [1] and Ladas and Lakshmikantham [6]. The author is grateful to T. G. Hallam for some useful comments on the preprint. 1. The general terminal comparison theorem. For b = +oo Theorem 1 below solves the problem explained in the introduction of the paper, while for b < +oo Theorem 1 improves Proposition A of Cafiero [2, p. 146] since no kind of boundedness is assumed on cA and the domain of co is more general. The proof Received by the editors July 10, 1974 and, in revised form, March 5, 1975. AMS (MOS) subject classifications (1970). Primary 34A40, 34A10, 34E99; Secondary 34A15, 34A45.

On the domain monotonicity of the first eigenvalue of a boundary value problem

Abstract: This paper is concerned with the monotonicity of the first eigenvalue λ1 (D) of (1) as a functional of the domainD.

Smoothness of the solution of a monotonic boundary value problem for a second order elliptic equation in a general convex domain

Abstract: We prove the square integrability of the second derivatives of the solution of an elliptic second order equation in a general convex domain, bounded in the n-dimensional Euclidean space, under monotonic boundary conditions. Our boundary conditions are general enough to include strongly non-linear conditions as for instance Signorini's. There is no restriction concerning the singularities of the boundary of the convex domain in which the equation is considered; this domain is allowed for instance to be a two-dimensional polygon or a three-dimensional polyhedron.

ON SPECTRAL DECOMPOSITIONS OF FUNCTIONS IN H_p^{\alpha}

Abstract: The paper is devoted to a study of the spectral resolutions and their Riesz means , corresponding to selfadjoint extensions of elliptic differential operators of order in an -dimensional domain . It is proved that if belongs to the Nikol'skiĭ class and has compact support in , then for , , and the Riesz means converge for to uniformly on each compact set . Bibliography: 9 titles.

Analytic Completion and Decomposablllty Properties In Tubold Domains

Abstract: The analytic structure of distributions -i.e. the decomposability of the latter in sums of boundary values of analytic functions can be investigated not only from the microlocal point of view, but more globally in complex space. For this purpose we introduce in Cn (or in more general manifolds) a large class of domains D called "tuboids", the closure of which contains a real domain J2 (J2 = D p)-Kn) -the boundary values on Q of functions which are analytic in D can then be considered (here in the sense of distributions ). It turns out that the decomposability properties in domains of this type are intimately related with the geometry of the latter, in particular with their pseudoconvexity properties. In this connection, we shall state some pseudoconvexity criterions and theorems of analytic completion.