Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Convergence of the highest derivatives in projection methods

Assume that for the approximate solution of an elliptic differential equation in a bounded domain , under a natural boundary condition, one applies the Galerkin method with polynomial coordinate functions. One gives sufficient conditions, imposed on the exact solutionu ^*, which ensure the convergence of the derivatives of order k of the approximate solutions, uniformly or in the mean in or in any interior subdomain. For example, ifu ^*W^k ₂, then the derivatives of order k converge in L₂(), where is an interior subdomain of . Somewhat weaker statements are obtained in the case of the Dirchlet problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 11–18, 1977.The author expresses his gratitude to Yu. K. Dem'yanovich for drawing his attention to [10].     

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.     

Nonlocal problems in the theory of the motion equations of Kelvin-Voight fluids

For the motion equations of Kelvin-Voight fluids one proves: 1) a global theorem for the existence and uniqueness of a solution (v;{u_e}) of the initial-boundary value problem on the semiaxis t R⁺ from the class W ¹ (R⁺); W ₂ ² () H()) with initial condition v_o(x) W ₂ ² () H() when the right-hand side f(x, t) L(R ⁺; L₂()); 2) a global theorem for the existence and uniqueness of a solution (v; {u_l}) on the entire axisR from the classW ¹ (R; W ₂ ² () H()) when the right-hand side f(x, t) L(R; L₂()); 3) a global theorem for the existence of at least one solution (v; {u_l}), periodic with respect to t with period , from the class W ¹ (R ⁺; W ₂ ² () H()) when the right-hand side f(x, t) L(R ⁺; L₂()) is periodic with respect to t with period , and a local uniqueness theorem for such a solution; 4) a theorem for the existence and uniqueness in the small of a solution (v; {u_l}), almost periodic with respect to t R, from V. V. Stepanov's class S ¹ (R; W ₂ ² ()H()) when the right-hand side f(x, t) S(R; L₂()) is almost periodic with respect to t; 5) the linearization principle (Lyapunov's first method) is justified in the theory of the exponential stability of the solutions of an initial-boundary value problem in the space H() and conditions are given for the exponential stability of a stationary and periodic solution, with respect to t R, of the system (1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 146–185, 1990.     

Some Non-trivial Symmetry Classes of Tensors Associated with Certain Characters

Let G be a finite group and a set of n elements. Assume that G acts faithfully on and let V be a vector space over the complex field , with dim V = m 2. It is shown that for each irreducible constituent of permutation character of G, the symmetry class of tensors associated with G and is non-trivial. This extends a result of Merris and Rashid (see [6, Theorem 2]).1995 AMS subject classification primary 20C30 secondary 15A69This research was in part supported by a grant from IPM.     

Functions with a finite dirichlet integral in a domain with cusp points on the boundary

Let be a domain in ⁿ, n >2, the boundary of which has a cusp point, pointing inside or outside the domain. The purpose of the paper is to characterize the traces on of the elements of the space H¹() of functions with a finite Dirichlet integral. As a consequence one establishes the existence of a linear continuous extension operator H¹ () H¹(ⁿ) under the presence of an interior cusp point on . Theorems on domains with cusps are proved with the aid of results on cylindrical domains. In the space of functions with a finite Dirichlet integral in the exterior or the interior of the cylinder one introduces the norm, depending on a small parameter and generating a norm of the trace on as an element of the quotient space. The latter is placed in correspondence with an explicitly described norm of functions on the boundary, uniformly equivalent relative to . One constructs an operator of extension of functions from the exterior of the cylinder to Rⁿ, preserving H¹, whose norm is uniformly bounded relative to . For the optimal operator of extension from the inside of the cylinder one finds the asymptotic behavior of the norm as 0. From these results there follow similar theorems on functions with a finite Dirichlet integral inside and outside a thin closed tube (of width ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 117–137, 1983.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

Homogenization of Some Elasticity Problems with Rapidly Oscillating Convex Constraints on Displacements

The problem of homogenization is considered for an elastic body occupying a perforated domain = obtained from a fixed domain and an -contraction of a 1-periodic domain .     

Relaxation of a Quadratic Functional Defined by a Nonnegative Unbounded Matrix

We study the lower semicontinuous envelope in L^p(), F, of a functional F of the form F(u)=A uudx where A=A(x) is not strictly elliptic and not bounded. We prove that F; may also be written as F;(u)= Buudx with B=AP A for a matrix P which is the matrix of an orthogonal projection. In the one-dimensional case, we characterize the domain of F and we explicit the matrix P.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains

We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W _m ¹( _s ) [W _m ¹( _s )]^* in a sequence of perforated domains _s . Under a certain condition imposed on the local capacity of the set \ _s , we prove the following principle of compensated compactness: , where r _s(x) and z _s(x) are sequences weakly convergent in W _m ¹() and such that r _s(x) is an analog of a corrector for a homogenization problem and z _s(x) is an arbitrary sequence from whose weak limit is equal to zero.     

Operatori lineariT-invarianti rispetto ad un gruppo di congruenze

Summary Let be an open subset of ⁿ, W_m() the linear space of m-vector valued functions defined on , G{} a group of orthogonal matrices mapping onto itself and T{T()} a linear representation of order m of G. A suitable groupC(G,T) of linear operators of W_m(), which leads to a general definition of T-invariant linear operator with respect to G, is here introduced. Characterization theorems concerning the linear differential and integral T-invariant operators are also given. When G is a finite group, projection operators are explicitly obtained; they define a «maximal» decomposition of W_m() into a direct sum of subspaces each of them invariant with respect to any T-invariant linear operator of W_m(). Some examples are givenc.

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Lavoro eseguito nell'ambito del progetto nazionale di ricerca «Analisi numerica e matematica computazionale» nell'anno 1985–86.     

Extensions from the Sobolev Spaces H 1 Satisfying Prescribed Dirichlet Boundary Conditions

Extensions from H ¹(_P) into H ¹() (where _P ) are constructed in such a way that extended functions satisfy prescribed boundary conditions on the boundary of . The corresponding extension operator is linear and bounded.     

Isoperimetric inequalities in a boundary value problem on a Riemannian manifold

In this paper the problem u+1=0 in ,u=0 on is considered. Here is a finite domain on a Riemannian manifold and the associated Laplace-Beltrami operator. By means of maximum principles isoperimetric bounds for the maximum ofu and the maximum of the absolute value of the gradient ofu, as well as some related bounds are derived.

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Zusammenfassung Diese Arbeit behandelt das Problem u+1=0 in ,u=0 auf , wobei ein Gebiet auf einer zweidimensionalen Riemann'schen Mannigfaltigkeit ist, und der zugehörige Laplace-Beltrami Operator. Es werden isoperimetrische Schranken für das Maximum vonu und |u| aus gewissen Maximumsprinzipien hergeleitet, sowie einige verwandte Resultate.

     

Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements

Summary A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain is partitioned into two subdomains ₁ and ₂.Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in . Finally, a hybrid situation in which viscosity is dropped out only in ₁ is addressed. The latter is motivated by physical applications.In all cases, correct transmission conditions across the interface between ₁ and ₂ are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed.The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out.We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size.This work was partially supported by CIRA S.p.A. under the contract Coupling of Euler and Navier-Stokes equations in hypersonic flowsDeceased     

On the problem of two linearized wells

We study the behaviour of sequences of elastic deformationsy ^{n n} whose gradients approach two linearized wells, and give an application to magnetostriction.This article was processed by the author using the style filepljour1m from Springer-Verlag.     

The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets

Using combinatorial and model-theoretic means, we examine the structure of normal subgroup lattices N(A()) of 2-transitive automorphism groups A() of infinite linearly ordered sets (, ). Certain natural sublattices of N(A()) are shown to be Stone algebras, and several first order properties of their dense and dually dense elements are characterized within the Dedekind-completion of (, ). As a consequence, A() has either precisely 5 or at least 2²¹ (even maximal) normal subgroups, and various other group- and lattice-theoretic results follow.     

A practical finite element approximation of a semi-definite Neumann problem on a curved domain

Summary This paper considers the finite element approximation of the semi-definite Neumann problem: –·(u)=f in a curved domain ⁿ (n=2 or 3), on and , a given constant, for dataf andg satisfying the compatibility condition . Due to perturbation of domain errors ( ^h ) the standard Galerkin approximation to the above problem may not have a solution. A remedy is to perturb the right hand side so that a discrete form of the compatibility condition holds. Using this approach we show that for a finite element space defined overD ^h , a union of elements, with approximation powerh ^k in theL ² norm and with dist (, ^h )Ch ^k , one obtains optimal rates of convergence in theH ¹ andL ² norms whether ^h is fitted ( ^h D ^h ) or unfitted ( ^h D ^h ) provided the numerical integration scheme has sufficient accuracy.Partially supported by the National Science Foundation, Grant #DMS-8501397, the Air Force Office of Scientific Research and the Office of Naval Research