Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

On a Singular Semilinear Elliptic Boundary Value Problem and the Boundary Harnack Principle

On a bounded C ²-domain we consider the singular boundary-value problem 1/2u=f(u) in D, u _D =, where d3, f:(0,)(0,) is a locally Hölder continuous function such that f(u) as u0 at the rate u ^–, for some (0,1), and is a non-negative continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in . Such solutions are shown to satisfy a boundary Harnack principle.     

Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let be a bounded domain in ⁿ (n3) having a smooth boundary, let be an essentially bounded real-valued function defined on × ^h, and let be a continuous real-valued function defined on a given subset Y of Y ^h. In this paper, the existence of strong solutions u W ^2,p (, ^h) W _o ^1,p (n/2<p<+) to the implicit elliptic equation (–u)=(x,u), with u=(u₁, u₂, ..., u_h) and u=(u ₁, u ₂, ..., u _h), is established. The abstract framework where the problem is placed is that of set-valued analysis.     

Positivity Preserving Forms have the Fatou Property

We prove that if (,D) is a positivity preserving form on L ² (E;m), and if (u _n)_n is a sequence in D() converging m-almost everywhere to u L ² (E;m), then (u,u) lim inf_n (u _n ,u _n ).     

Sur une Classe de Fluides Non-Newtoniens

A constitutive law for a class of non newtonian fluids is considered. The stress-tension is defined as an element in the subgradient of a convex, l.s.c. function: (D);D is the tension of the rate of deformation. We give existence and uniqueness theorems. Some examples (Bingham fluid, pseudo-plastic and dilatant fluids) are also given.     

Analogues of hadwiger's theorem in space ℝ n —Sufficient conditions for a convex domain to enclose another

We estimate the kinematic measure of one convex domain moving to another under the groupG of rigid motions in ⁿ . We first estimate the kinematic formula for the total scalar curvature D ₀gD ₁ Rdv of then–2 dimensional intersection submanifold D ₀gD ₁. Then we use Chern and Yen's kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in ⁿ (4). Forn=4, we directly obtain another sufficient condition in ⁴.     

On Fine Limits and Curving-Shaped Limits

This paper studies the boundary behavior of the so-called SIH-functions, i.e., the functions satisfying the scale invariant Harnack inequality on a domain D R^N (N 2). Suppose that D contains a curving-cone at a point D and u is a SIH-function on D. Then u has a curving-shaped limit L at , if u has a -fine limit (especially, a p-fine limit in the sense of [6] or an -fine limit in the sense of the Riesz potential theory [4]) L at .AMS Subject Classification (2000): 31B25, 31C15, 30C65     

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Summary The Skorohod oblique reflection problem for (D, , w) (D a general domain in ^d , (x),xD, a convex cone of directions of reflection,w a function inD(⁺, ^d )) is considered. It is first proved, under a condition on (D, ), corresponding to (x) not being simultaneously too large and too much skewed with respect to D, that given a sequence {w ⁿ} of functions converging in the Skorohod topology tow, any sequence {(x ⁿ, ⁿ)} of solutions to the Skorohod problem for (D, , w ⁿ) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, , w). Next it is shown that if (D, ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, , w) exist for everywD(⁺, ^d ) with small enough jump size. The requirement is met in the case when D is piecewiseC _b ¹ , is generated by continuous vector fields on the faces ofD and (x) makes and angle (in a suitable sense) of less than /2 with the cone of inward normals atD, for everyxD. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.     

Approximation of a nondifferentiable nonlinear problem related to MHD equilibria

Summary We present a simple method, based on a variant of the implicit function theorem, which leads to the existence of (a part of) a nontrivial solution branch of the nonlinear eigenvalue problem –u=u ⁺ in ,u=–1 on , where is a two-dimensional domain with boundary . The advantage of this method is that we can apply it for analysing the approximation of the above problem by a finite element method; the error analysis of the discrete problem appears immediately. We give also an iteration scheme which allows to solve the approximate problem.     

Symmetry and non-symmetry for the overdetermined Stekloff eigenvalue problem

We consider the Stekloff eigenvalue problem (1.1)–(1.2); Payne and Philippin conjectured that ifu is an eigenfunction which satisfies the overdetermined condition u=1 on , then should be a disk. In this paper we show that this conjecture holds if and only if the complex potentialF associated tou vanishes only at one point. Then we show how to construct non-symmetric domains in the case whereF vanishes at more than one point.     

On boundary controllability of a vibrating plate

A vibrating plate is here taken to satisfy the model equation:u _tt + ²u = 0 (where ²u:= (u); = Laplacian) with boundary conditions of the form:u _v = 0 and(u) _v = = control. Thus, the state is the pair [u, u _t] and controllability means existence of on := (0,T)× transfering any[u, u _t]₀ to any[u, u _t]_T. The formulation is given by eigenfunction expansion and duality. The substantive results apply to a rectangular plate. For largeT one has such controllability with = O(T ^–1/2). More surprising is that (based on a harmonic analysis estimate [11]) one has controllability for arbitrarily short times (in contrast to the wave equation:u _tt = u) with log = O(T ^–1) asT0. Some related results on minimum time control are also included.This research was partially supported under the grant AFOSR-82-0271.     

Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ɛ-subdifferential of the associated convex quadratic form

LetA(·) be ann × n symmetric affine matrix-valued function of a parameteruR ^m , and let (u) be the greatest eigenvalue ofA(u). Recently, there has been interest in calculating (u), the subdifferential of atu, which is useful for both the construction of efficient algorithms for the minimization of (u) and the sensitivity analysis of (u), namely, the perturbation theory of (u). In this paper, more generally, we investigate the Legendre-Fenchel conjugate function of (·) and the -subdifferential (u) of atu. Then, we discuss relations between the set (u) and some perturbation bounds for (u).The author is deeply indebted to Professor J. B. Hiriart-Urruty who suggested this study and provided helpful advice and constant encouragement. The author also thanks the referees and the editors for their substantial help in the improvement of this paper.     

Propriété de stabilité de la fonction extrémale relative

Nous donnons une caractérisation des domaines DX pour lesquels la fonction extrémale relative ^*(,E,D) a la propriété de stabilité pour tout ED, i.e. lim _k^*(,E,D _k )=^*(,E,D), ED. Ensuite, nous étudions la relation entre cette propriété et les enveloppes pluripolaires. Nous concluons par quelques remarques sur la propriété de stabilité lim _k^*(,E _k ,D)=^*(,E,D).     

On certain imbedding and extension theorems for a weighted class of functions

n- , S n–1 m (0m<n), () S . () , ().     

Note on the effectiveness factor of a first order reaction

Upper and lower bounds for the effectiveness factor are derived, whereu is the solution of u=c ² u in,u=1 on .

---

Zusammenfassung Es werden obere und untere Schranken hergeleitet für den Diffusionsfaktor , wobeiu die Lösung ist von u=c ² u in,u=1 auf .

---

---

This research was done while the author was visiting Cornell University, Ithaca, New York.     

Solutions of Δ<Emphasis Type="Italic">u</Emphasis>=4<Emphasis Type="Italic">u</Emphasis><Superscript><Emphasis Type="Italic">2</Emphasis></Superscript> with Neumann’s conditions using the Brownian snake

We consider a Brownian snake (W_s,s0) with underlying process a reflected Brownian motion in a bounded domain D. We construct a continuous additive functional (L_s,s0) of the Brownian snake which counts the time spent by the end points _s of the Brownian snake paths on D. The random measure Z=_sdLs is supported by D. Then we represent the solution v of u=4u² in D with weak Neumann boundary condition 0 by using exponential moment of (Z,) under the excursion measure of the Brownian snake. We then derive an integral equation for v. For small it is then possible to describe negative solution of u=4u² in D with weak Neumann boundary condition . In contrast to the exit measure of the Brownian snake out of D, the measure Z is more regular. In particular we show it is absolutely continuous with respect to the surface measure on D for dimension 2 and 3.Mathematics Subject Classification (2000):60J55, 60J80, 60H30, 60G57, 35C15, 35J65     

Parabolische Kollineationen,die eine ebene konvexe Menge invariant lassen

Collineations ₁, ₂ of PG(2, ) leaving invariant a compact convex setK ² are called parabolic if |K Fix _i|=1. Conditions are stated under which the existence of ₁, ₂ imply that K is an ellipse.

Herrn Helmut R. Salzmann zum 65. Geburtstag gewidmet     

Minimization of the Conformal Radius under Circular Cutting of a Domain

Let D be a simply connected domain on the complex plane such that 0 D. For r > 0 , let D _r be the connected component of D {z : |z| < r} containing the origin. For fixed r, we solve the problem on minimization of the conformal radius R(D _r, 0) among all domains D with given conformal radius R(D, 0). This also leads to the solution of the problem on maximization of the logarithmic capacity of the local -extension E (a) of E among all continua E with given logarithmic capacity. Here, E (a) = E {z : |z – a| }, a E, > 0. Bibliography: 12 titles.     

Addendum to ‘asymptotic eigenfunctions of mixed problems of Stekloff's type’: Extension to exterior problems

Summary The asymptotic eigenvalues and eigenfunctions of the problem: V=0 in the exterior domain to a regular closed curvec and V/=V(s)g(s) onc, are obtained. It is shown that _n =0(n) andV _n (s) becomes trigonometric.

---

Résumé SoitD la région extérieure d'une courbe fermée régulière,c; on détermine le comportement asymptotique des valeurs propres et fonctions propres du problème: V=0 dansD et V/=V(s)g(s) surc. On peut démontrer que _n =0(n) et lesV _n (s) deviennent des fonctions trigonométriques.

     

On the Uniqueness ofL 2-solutions in half-space of certain differential equations

Square integrable solutions to the equation{– ²/y² + P(D_x)+b(y)–}u(x, y) = f(x, y) are considered in the half-spacey>0, x ⁿ , whereP(D _x) is a constant coefficient operator. Under suitable conditions on lim_y0u(x, y), b(y), f(x, y) and , it is shown that suppu = suppf. This generalizes a result due to Walter Littman.Research partially supported by USNSF Grant 79-02538-A02.