Showing papers on "Domain (mathematical analysis) published in 1973"

Uniform estimates for the\(\bar \partial \)-equation on domains with piecewise smooth strictly pseudoconvex boundaries

Abstract: Henkin [3] and Grauert-Lieb I7] pioneered in the investigation of uniform estimates for the ~-equation. They independently proved the following: if D is a strictly pseudoconvex domain in fEn with smooth boundary and i f f is a uniformly bounded R-closed C °~ (0, 1)-form on D, then there exists a uniformly bounded C ° function u on D with du = f. Their proofs depend on the explicit construction of a solution by means of holomorphic kernels introduced by Henkin [2] and Ramirez [19]. Later, by using a local version of the Grauert-Lieb method, Kerzman 1'9] generalized the result to a strictly pseudoconvex domain with smooth boundary in a Stein manifold. Moreover, he showed that the type of solution constructed by Grauert-Lieb for such a domain yields also LP-estimates and H61der estimates with exponents 1/2. By modifying Henkin's solution of the ~-equation, Henkin and Romanov 1'5] obtained the H61der estimate with exponent t/2. Making use of Koppelman's results 1'1 t], Lieb [13, 14] proved that there exist uniform estimates and H/ilder estimates with exponent < 1/2 for the ~-equation for (0, q)-forms on strictly pseudoconvex domains with smooth boundary. Independently ~vrelid [16] also obtained uniform estimates and L p estimates for the ~-equation for (0, q)-forms on the same class of domains. Recently Henkin 1'4] announced the solution of the R-equation with uniform bounds for (0, 1)-forms for certain analytic polyhedra and indicated that his proof makes use of the type of Cauchy-Fantappie kernel introduced by Norguet 1'15]. In this paper we investigate uniform estimates for solutions of the R-equation on a domain with piecewise smooth strictly pseudoconvex boundary. More precisely, suppose D is a bounded domain in C n with aD covered by finitely many open subsets Uj(l _~j_~ k) ofC n and suppose 01 is a ¢2 strictly plurisubharmonic function on UI(1 < j < k) such that

Contour and solid structure properties of holomorphic functions of a complex variable

Abstract: For a function holomorphic in an open set the paper solves problems on the relationships between its properties along , the boundary of , on the one hand and along , the closure of , on the other. The properties discussed are those that can be expressed in terms of the derivatives, moduli of continuity, and rates of decrease or increase of the function along and along . The results are established for very wide classes of sets and majorants of the moduli of continuity. In particular, all the main results are true for every bounded simply-connected domain and any majorant of the type of a modulus of continuity. A number of problems posed in 1942 by Sewell are solved.

Asymptotic stability and spiraling properties for solutions of stochastic equations

Abstract: We consider a system of Ito equations in a domain in Rd. The boundary consists of points and closed surfaces. The coefficients are such that, starting for the exterior of the domain, the process stays in the exterior. We give sufficient conditions to ensure that the process converges to the boundary when t-'o. In the case of plane domains, we give conditions to ensure that the process "spirals"; the angle obeys the strong law of large numbers. Introduction. In a previous work [4] we have investigated the behavior of solutions of linear stochastic differential equations when t -oo. The purpose of the present work is to extend the results of [4] to nonlinear equations. Specifically we shall consider a Markov process on R 1 defined by the stochastic equations n dx. a ci(x) dws + b (x) dt (1 < i < 1, 1 < s < n), s=1 Xi(O)-=Xi = xi together with a "stable manifold" dG. The set G will consist of a finite number of points together with a finite number of closed domains. The coefficients ais b are such that if the process starts on dG then it stays forever on dG. Our first result (Theorem 1.1) gives a set of sufficient conditions for the nonattainability of G, starting from the exterior. If G consists of points and convex bodies, it suffices that the normal cornponents of the diffusion and the drift vanish on dG; in general we need to impose an additional "convexity" relation between dG, the drift and the diffusion coefficients to ensure the nonattainability of dG. The next result (Theorem 2.1) gives sufficient conditions that x(t) -a dG when t -* oo. This theorem contains local stability conditions (near dG and near oo) reminiscent of the linear case [4], as well as a certain nondegeneracy condition. None of these conditions can be relaxed. Received by the editors May 22, 1972 and, in revised form, December 11, 1972. AMS (MOS) subject classifications (1970). Primary 60H10, 60J60; Secondary 34D05, 34C05.

Polygonal Domain Approximation in Dirichlet's Problem

Abstract: : Consider Dirichlet's problem with vanishing boundary values for Poisson's equation in a smooth domain omega in the plane. In the paper upper and lower bounds with respect to different norms are given for the error and the gradient of the error introduced by approximating omega by a polygonal domain omega sub h with side-lengths at most h. (Author)

A remark on level curves for domains convex in one direction

Abstract: Let D be a plane domain which is convex in the v-direction, i.e.t the intersection of D with each vertical line is connected (or empty). It has been an open question whether level curves of a domain convex in the v-direction bound a domain with the same property. In this note we construct an example which settles the question in the negative.Closely related is a family ? of analytic functions g in the unit disk with the property that g(0) = 0 and Re{(1 - z2)g(z)/z} ⋚ 0. For univalent functions we show that membership in ? is essentially characterized by the geometric condition that Im g(ei0)⋚ 0 for a.e. ? ? (0, ?) and Im g(ei0)⋚ 0 for a.e. We conclude with a coefficient theorem

On polynomial approximation in _

Abstract: Let D be a bounded Jordan domain with rectifiable boundary and define AJ(D), the Bers space, as the space of holomorphic functionsf, such that ffif I2D dxdy D is finite, where AD is the Poincare metric for D. It is shown that the polynomials are dense in AJ(D) for q>3/2.

Some recent advances in the mathematics of finite elements

Abstract: Publisher Summary This chapter discusses the mathematical theory of finite element methods. It also discusses the Sobolev spaces and variational formulation of elliptic boundary value problems, nodal finite element method, abstract finite element method, and nonlinear elliptic problems and time dependent problems. It is well known that the finite element method is a special case of the Ritz–Galerkin method. The classical Ritz approach has two great shortcomings: (1) in practice, construction of the basis functions is only possible for some special domains: (2) the corresponding Ritz matrices are full matrices, and are very often, for simple problems, catastrophically ill-conditioned. The crucial difference between the finite element method and the classical Ritz–Galerkin technique lie in the construction of the basis functions. In the finite element method, the basis functions for general domains can easily be computed. The main feature of these basis functions is that they vanish over all but a fixed number of the elements into which the given domain is divided. This property causes the Ritz matrices to be sparse band matrices, and the resulting Ritz process is stable. There are in addition to the Ritz–Galerkin method many other direct variational methods, and the differences among these stem from the variational principles used.

Nonlinear Equation of Unsteady Ground-Water Flow

Abstract: A finite element weighted residual process has been used to solve nonlinear partial differential equations describing unsteady ground-water flows in an unconfined aquifer either into or out of a surface reservoir. Rectangular, as well as triangular, finite elements in a space-time solution domain were used. The weighting function was equal to the shape function defining the dependent variable approximation. The results are compared in dimensionless graphs with experimental as well as other numerical data. The finite element method compared favorably with these results and was found to be easily programmed, stable, computationally fast, rapidly convergent, and does not require constant parameters over the entire solution domain. This technique is another useful tool in solving field problems.

On convolution equations I

Abstract: Various additive formulae for supports and singular supports of convolutions are studied in terms of the Fourier transform in complex domain. Part of the material presented below was announced in our note [4]. The authors with to thank Professor L. Hormander for his most hepful comments.

Boundedly holomorphic convex domains

Abstract: shown in this paper: In a Riemann domain, a boundedly holomorphic convex domain is a domain of bounded holomorphy. With some restrictions, the converse is true. The spectrum of the algebra B of bounded holomorphic functions is an envelope of bounded holomorphy provided that the completion of B with the topology of uniform convergence on compact subsets is stable under differentiation. Finally, Stein manifolds of bounded type are introduced. Let (Xl9 Aj) and (Xi9 A2) be complex (analytic) manifolds. A map oi\ Xx —> X2 said to be biholomorphίc if a: is a homeomorphism of Xλ onto X2 and both a and or1 are holomorphic. a is called a spread map if a is a locally biholomorphic. We denote a complex manifold (X, A; a) with a spread map a. A Riemann domain is a complex manifold which spreads in (C*, <£?). We denote B{X) for the algebra of all bounded holomorphic functions on X. DEFINITION l Let (X, A) be a complex manifold and D be open in X. Let B = B(D). D is said to be boundedly holomorphic convex if Ίcm\\BK= KB = {xeD; \f(x)\ ^ \\f\\κ for all feB] is compact provided if is a compact subset of D. An open set D of X is called a region of bounded holomorphy if there is an / e B(D) for which every boundary point of D is a boundary singularity in the sense that / has no bounded analytic continuation to any open neighborhood of any boundary point (see [5]). The following natural questions arise; if boundedly holomorphic convex domains are domains of bounded holomorphy, and vice versa. The answer for the first is affirmative.

Concave solutions of a Dirichlet problem

Abstract: We define a function that may serve as the measure of the deviation of a plane convex domain from circular shape. Then we show that the Dirichlet problem of Au+Ap(x, y)u=O and vanishing boundary values on a plane, convex domain can have a concave solution only if an integral condition involving the deviation from circular shape is satisfied. A weak form of the condition is generalized to n dimensions. The basic estimate for the first eigenvalue of a problem x" + 2,p(t)x = 0, p(t) > 0 x(a) = x(b) = 0, is Ljapunov's inequality rb (1) Xf p(t) dt ? 4/(b -a) The estimate has been generalized to p(t) of varying sign and to partial differential equations. Joukovsky [4] has given an elegant proof of (1) based on the fact that the graph of the first eigenfunction together with the segment [a, b] bounds a convex domain. In this note, we use a method inspired by that of Joukovsky to study concave solutions of Au + Ap(x)u = 0, p(x) _ 0, defined in a convex domain D of real n-space of coordinates x= (x,... *, x1,) for vanislhing boundary values, UIaD=0. For given p, the behavior of the solution obviously depends on the geometry of D. The important features can be studied on the example of the equation with constant coefficients, Au + Xu=O in D, u = O on aD. Received by the editors September 15, 1972 and, in revised form, December 18, 1972. AMS (MOS) stibject classifications (1970). Primary 35J25, 52A10; Secondary 35P15, 52A20. I Research partially supported by NSF Grant GP-27960. ? American Mathematical Society 1973

On the Approximate Solution of the First Boundary Value Problem for $\nabla ^4 u = f$

Abstract: For the equation $ abla ^4 u = f$ in a bounded domain, $\Omega \subset R^n $, with infinitely smooth boundary $\Gamma $, it is shown that the solution u to the boundary value problem which prescribes the function and its normal derivative on $\Gamma $ may be approximated as closely as desired by solving a sequence of Dirichlet problems for Poisson equations in the same domain, provided that the data is sufficiently regular.

The First Boundary Value Problem

Abstract: We shall denote by Ω a bounded domain in Euclidean space R m , and by x = (x1, ⋯, x m ) a point in this space. All functions considered in Chapter I are assumed real unless otherwise specified. As usual, the symbol A ⊂ B means that the set A is contained in B, and Ā denotes the closure of the set A.

The neumann eigenvalue problem

Abstract: Let A be a bounded domain of the z-plane (z = x+i y) with a piece-wise smoorh boundary. In this paper the following elgenvalue problem will be considered denotes differentiation along the normal to )

Value Distribution of Axisymmetric Potentials

Abstract: where g is an analytic function referred to as the associate of *. The domain of g, W, is an axiconvex subset of the complex plane, C, meaning that whenever g is contained in w then the entire line segment joining g and C is in o. The domain of j is the axisymmetric region obtained by rotating o abolut the real axis. Marden (4, p. 142) employs methods drawn from the analytic theory of polynomials in one complex variable to study A. S. P. and G. A. S. P. whose associates are rational functions. In particular. the set of points in C where the rational function assumes an assigned value is used to determine a pair of cones in E3 and more generally El, n > 4, where the corresponding A. S. P. or G. A. S. P. omits this value.

Locality conditions on form factors

Abstract: We derive a set of integral equations which are necessary and sufficient conditions on the form factors of local field theory, i.e. on the matrix elements of local operators. The basic idea is that out of the set of all (distribution‐valued) functions defined on the boundary of an analyticity domain, in general only a subset are boundary values of functions which are analytic within the domain. The form factors are boundary values of a vertex function which, due to the general assumptions of locality, reasonable energy, and mass spectrum and Poincare covariance, is analytic at least in the domain constructed by Kallen and Wightman. The characteristic boundary of the domain (``the distinguished boundary'') is the set of physical values of the arguments of the form factors, and the integral equations in that way only involve such values. The main advantages in formulating the locality conditions in this way are that (1) only the physical quantities of the field theory, i.e., the matrix elements between the field operators, enter into the equations and (2) the frustrating complications which are met in the construction of the domains of analyticity for n‐point functions with n > 3 might hopefully be avoided because the distinguished boundary can be constructed even if the whole domain is not known. The integral equations have naturally no unique solutions, because, e.g., all perturbation‐theoretical form factors must, of course, fulfill them. The equations may, however, function as a convenient starting point for approximations and ``model building'' for form factors outside the presently used perturbation theories. The integral equations are straightforward generalizations of the notion of ``weak local commutativity'' for the two‐point function. This condition means that the two spectral functions connected to two locally commuting operators should be equal. The conditions on the form factors (which are the generalizations of the two‐point spectral functions) are that the difference should vanish when integrated over certain physical sets of mass space.

A sufficient condition for global constrained extrema

Abstract: Sufficient conditions for global constrained minima and for global minima in some restricted domain of the problem are given. Both conditions express the fact that the Lagrangian function is convex, which does not require any convexity property of the functional and the constraints.

An elementary proof of a theorem concerning infinitely connected domains

Abstract: Using classical complex function theory, it is shown that any infinitely connected plane domain is conformally equivalent to a domain whose isolated boundary components are analytic Jordan curves. This allows an elementary proof to be given of the result that a domain with countably many boundary components is conformally equivalent to a domain bounded by analytic Jordan curves. 1. It is an immediate consequence of the Riemann mapping theorem that any domain of,finite connectivity can be mapped conformally onto a domain whose boundary components are analytic Jordan curves. In recent work ([2], [3]) the author has shown that this result holds for infinitely connected domains (with countably many boundary components). The proof, by transfinite induction, uses only the standard theory of normal families and the following: every plane domain is conformally equivalent to a domain whose isolated boundary components are analytic Jordan curves. (Points are to be considered as degenerate analytic Jordan curves, but in any case if they are isolated they can be ignored.) The proofs given in [2] and [3] of this statement use deep results in the theory of quasiconformal mappings. We give here a direct "normal family proof" (a simplified version of the transfinite induction argument in [3]). Once this is done, the entire proof of the general theorem (see ?3) becomes elementary-using only classical function theory (cf. [3, Note, p. 418]). 2. We recall the classical proof for finitely connected domains. If the boundary components of such a domain Do are indexed by k=-, * , * , n and this indexing is preserved under conformal maps of the domain, then one obtains maps fk (O_k?n) of Do onto domains Dk whose first k boundary components are analytic Jordan curves. This is done by successively applying the Riemann mapping theorem to the domain bounded Received by the editors March 31, 1972. AMS (MOS) subject class{/lcations (1970). Primary 30A30.

The local range set of a meromorphic function

Abstract: Let f be a function meromorphic in a domain G of the Riemann sphere. The global range set of f is the set of values assumed infinitely often byf, and similarly the local range set off at a boundary pointp is the set of values assumed infinitely often in every neighborhood of p. Obviously any range set is a G,6 set. In this paper we show that every G, set is the local range set of some meromorphic function. This contrasts with the situation for the global range set. Our methods rely on prime end theory and Arak6lian's approximation theorems. Following Rudin [9], we call a set V a A-set if V is a countable intersection of nested domains. In case G is the unit disc, then the local and global range sets are A-sets, and conversely it is known that every A-set is the (local and global) range set of some function meromorphic in the unit disc (see [7], [9]). Recently, it has been established that for an arbitrary domain, the global range set is always a A-set [3, p. 670]. Thus if the closure of the global range set has a nonempty interior then the linear Hausdorff measure is infinite, in fact it is not even a-finite [3, p. 670]. In this paper we study the local range set. In ?1 we extend Arakelian's approximation theorems. In ?2 we examine simply connected domains and find a type of boundary point at which the local range set is always a A-set, and a type of boundary point at which the local range set is not necessarily a A-set. Finally we exhibit a domain having a boundary point at which any G6 set can be realized as the local range set of some meromorphic function. Note that there are G6 sets which are dense in the plane and of linear Hausdorff measure zero. We shall make use of the following notations and definitions. If G is a proper domain of the Riemann sphere, then G* denotes the onepoint compactification of G and infinity will always stand for the ideal Received by the editors January 15, 1973. AMS (MOS) subject class'ifcations (1970). Primary 30A72; Secondary 30A82.

Improperly Posed Initial Value Problems for Self-Adjoint Hyperbolic and Elliptic Equations

Abstract: Integral representations are obtained for the solution to Cauchy’s problem for hyperbolic equations along a convex time-like surface, the exterior characteristic initial value problem for hyperbolic equations, and Cauchy’s problem for elliptic equations along an analytic surface. Each of these problems is improperly posed in the real domain and hence our representations are constructed by integrating over appropriate regions in the space of one and several complex variables.

Area Theorems for Functions Analytic in a Finitely Connected Domain

Abstract: In this paper we consider functions analytic in a given finitely connected domain apart from a finite number of singularities of possibly logarithmic type. We prove some area theorems which generalize to these functions certain known results, in particular Goluzin's area theorem on functions p-valent in a disc. We establish some integral criteria of when functions meromorphic in a given multiply-connected domain are univalent functions there and pairwise without common values.

On the Approximation of Solutions of Boundary Value Problems in Domains with AN Unbounded Boundary

Abstract: Elliptic problems with a complex parameter q are considered for equations with variable coefficients in domains Ω with an unbounded boundary. It is proved that for sufficiently large q the problem has a unique solution in the space HS(Ω) and that the solution can be obtained as the limit as r→∞ of the solution of a boundary value problem in a certain bounded domain Ωr⊂Ω. Bibliography: 6 items.

Unitary Variational Approximations

Abstract: The problem of generating nonrelativistic three-body scattering amplitudes which satisfy unitarity exactly, at all energies, is studied in the context of an effective-potential theory. It is shown how the nonlinear unitarity relations can be replaced by linear integral equations, simpler in structure than the original Faddeev equations, such that for any set of input amplitudes satisfying standing-wave boundary conditions the output will be unitary. Variational principles for these input amplitudes are derived from the Faddeev equations which define them, so that approximations can be systematically improved. Attention is drawn to a particular class of approximations for which the integral equations to be solved are all of the twobody type. These approximations have the additional virtue that the trial functions which enter into the Schwinger-type variational expression are square-integrable. This property allows for a choice of trial functions based on an "effective-range" type of argument. The Schwinger principle can then be though of as providing an analytic continuation of the effective potential from an energy domain below the breakup threshold to a limited range above it.

Two types of hyperinvariant subspaces

Abstract: Let A be a bounded operator in a Banach space B. Suppose that A has the single valued extension property. Given a closed set Fin the complexes, define rA(F) to be the set of all x in B such that there is an analytic function x(A) from the complement of F to B with (A -Al)x(A)=x. A is said to have property Q if 0A(F) is a closed subset of B for every F. Let A be, again, a bounded operator in a Banach space B. Given a real number b, define SA(b) to be the set of all x in B such that exp(-ct)exp(At)x is a bounded function from the nonnegative reals to B for all c>b. A is said to have property P if SA(b) is a closed subspace of B for all b. These two properties are discussed in this paper. Two types of closed invariant subspaces for a bounded operator A are the subject of this paper. One is related to the solution of inhomogeneous equations; the other is related to the asymptotic behavior of exp(At). Both are hyperinvariant or, in other words, invariant under all operators commuting with A. We define two properties related to these types of invariant subspaces, which guarantee that if they occur they are closed. Property Q holds for all decomposable operators (see Colojoara and Foias [1]) and has been long known as one of the properties possessed by spectral operators. In the sequel, A will be taken to be a bounded linear operator from a Banach space B into itself. We first define property Q. Suppose that A has the single valued extension property, or in other words that there is no solution x(A) of the equation (A-1I)x(Z)=0 for all A in some complex domain, such that x(A) is an analytic function from the domain to B. Define OA(X) to be the set of all AO in the complexes such that the equation (A AI)x(Q)=x is not solvable in any neighborhood of 2A, with x(A) analytic. For a closed set F in the complexes, define cA(F) to be the set of all x in B such that OA(x) does not intersect the complement of F. We say that A has property Q if aA(F) is closed, for every closed set F, and A has the single valued extension property. It is obvious that crA(F) is a hyperinvariant subspace. Received by the editors January 6, 1972. AMS (MOS) subject classifcations (1970). Primary 47A15.

Upper bounds for the fundamental eigenvalue for a domain of unknown shape

Abstract: For a particular eigenvalue problem in partial differential equations, upper bounds are established which do not depend on the shape of the domain but only on its size. The problem describes the free sloshing motions of an incompressible, inviscid fluid in a canal and furnishes upper bounds to the highest fundamental sloshing frequency which is attainable for a given cross-section area of the canal.

The inverse problem of approximating functions on a boundary for compacta of positive capacity

Abstract: in [i] were discussed a compactum B, the complement of which to the extended plane consists of a finite number of domains, and a function f(z), defined on the boundary L of B and approximated on L by rational functions with poles at fixed points (one pole in each complementary domain of B). For a given rate of approach in [1] were established bounds for the boundary* moduli of continuity of f(z) and its boundary derivatives, subject to the condition that the complement of B is regular in the sense that the Dirichlet problem can be solved for any continuous boundary values (ef. also [2-5]).