Dirichlet and Neumann boundary conditions: What is in between?

Given an admissible measure μ on where is an open set, we define a realization of the Laplacian in with general Robin boundary conditions and we show that generates a holomorphic C ₀ -semigroup on which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity assumption, the generator of each sandwiched semigroup is of the form . We also show that if contains smooth functions, then μ is of the form (where σ is the (n-1)-dimensional Hausdorff measure and β a positive measurable bounded function on ); i.e. we have the classical Robin boundary conditions. RID="h1" ID="h1"Dédié à Philippe Bénilan RID="*" ID="h1"This work is part of the DGF-Project: "Regularit?t und Asymptotik für elliptische und parabolische Probleme".     

Γ-Limit of Periodic Obstacles

We compute the -limit of a sequence obstacle functionals in the case of periodic obstacles.     

On zeros of polynomials orthogonal over a convex domain

We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal on a convex domain.     

Polynomial inequalities on measurable sets and their applications

We study Pólya-and Remez-type inequalities for univariate and multivariate polynomials and discuss their applications to Nikolskii-type inequalities and upper estimates of trigonometric integrals.     

Periodic points of positive linear operators and Perron-Frobenius operators

LetC(S) denote the Banach space of continuous, real-valued mapsf:S and letA denote a positive linear map ofC(S) into itself. We give necessary conditions that the operatorA have a strictly positive periodic point of minimal periodm. Under mild compactness conditions on the operatorA, we prove that these necessary conditions are also sufficient to guarantee existence of a strictly positive periodic point of minimal periodm. We study a class of Perron-Frobenius operators defined by

Perturbation of orthonormal bases inL 2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionsh _n that imply that the sequence (1+h _n (x))e ^inx is a Riesz basis forL ²[–,]. These conditions involve theL ²-norm of the second derivativesh _n . The new results replace the differential operatoryy by more general differential operators inL ²-spaces, in particular, by the Hermite differential operator inL ²(R), ande ^inx by arbitrary orthonormal bases.     

Banach space duality of absolute Schur algebras

Matrix Schur product is the entry-wise product of matrices of the same size. It was shown by P. Chaisuriya and S.-C. Ong [1] that (forr1) infinite matrices [a _jk ] such that [|a _jk | ^r ] B(l ² form a Banach algebra under the norm [a _jk ] _r =[|a _jk | ^r ]^1/r and the Schur product. In this paper we demonstrate the existence of Banach space duality within the class of these algebras which is analogous to the classical duality between the spaces of compact, trace class, and bounded operators onl ². Also we obtain a general functional calculus on these algebras, which is used to determine the spectrum and to justify the notion of -norm introduced in [1].This author's research was supported by a sabbatical leave and a Division grant from Colby College.This author's research was supported by a research professorship at Central Michigan University.This author's research was supported by a visiting professorship at Central Michigan University.     

Local solvability of partial differential operators on the Heisenberg group

In this article the following class of partial differential operators is examined for local solvability: Let P(X, Y) be a homogeneous polynomial of degree n ≥ 2 in the non-commuting variables X and Y. Suppose that the complex polynomial P(iz, 1) has distinct roots and that P(z, 0) = zⁿ. The operators which we investigate are of the form P(X, Y) where X = δ_x and Y = δ_y + xδ_w for variables (x, y, w) ∈ ?³. We find that the operators P (X, Y) are locally solvable if and only if the kernels of the ordinary differential operators P(iδ_x, ± x)* contain no Schwartz-class functions other than the zero function. The proof of this theorem involves the construction of a parametrix along with invariance properties of Heisenberg group operators and the application of Sobolev-space inequalities by Hörmander as necessary conditions for local solvability.     

Another Look at the Erdős–Hajnal–Pósa Results on Partitioning Edges of the Rado Graph

Dedicated to the memory of Paul Erdős Erdős, Hajnal and Pósa exhibited in [1] a partition (U,D) of the edges of the Rado graph which is a counterexample to . They also obtained that if every vertex of a graph has either in or in the complement of finite degree then . We will characterize all graphs so that . Received October 29, 1999 RID="†" ID="†" Supported by NSERC of Canada Grant #691325.