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1.
BARRETT JOHN W.; CHAKRABARTI ROMA; ELLIOTT CHARLES M. 《IMA Journal of Numerical Analysis》1991,11(4):579-594
Optimal order H1 and L error bounds are obtained for a continuouspiecewise linear finite element approximation of an obstacleproblem, where the obstacle's height as well as the contactzone, c, are a priori unknown. The problem models the indentationof a membrane by a rigid punch. For R2, given ,g R+ and an obstacle defined over E we consider the minimization of |v|21,+gµover (v, µ) H10() x R subject to v+µ on E. In additionwe show under certain nondegeneracy conditions that dist (c,hc)Ch ln 1/h, where hc is the finite element approximation toc. Finally we show that the resulting algebraic problem canbe solved using a projected SOR algorithm. 相似文献
2.
Logarithmic Convexity for Supremum Norms of Harmonic Functions 总被引:1,自引:0,他引:1
We prove the following convexity property for supremum normsof harmonic functions. Let be a domain in Rn, 0 and E a subdomainand a compact sebset of ,respectively. Then there exists a constant = (E, 0, ) (0, 1) such that for all harmonic functions u on, the inequality
is valid.The case of concentric balls E plays a key role in the proof.For positive harmonic funcitons ono osuch balls, we determinethe sharp constant in the inequlity. 相似文献
3.
Let be a bounded connected open set in RN, N 2, and let 0be the Dirichlet Laplacian defined in L2(). Let > 0 be thesmallest eigenvalue of , and let > 0 be its correspondingeigenfunction, normalized by ||||2 = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying
Let 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||2=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying
1991 Mathematics SubjectClassification 47F05. 相似文献
4.
Packing, Tiling, Orthogonality and Completeness 总被引:3,自引:0,他引:3
Let Rd be an open set of measure 1. An open set DRd is calleda tight orthogonal packing region for if DDdoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of Rd. The main contribution of this paperis a new way of proving the following result: D tiles Rd whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L2(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in Rd, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about packing Rd withtranslates of a certain non-negative function and, additionally,we have completeness of E in L2() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70. 相似文献
5.
Let G be a transitive permutation group on a set such that,for , the stabiliser G induces on each of its orbits in \{}a primitive permutation group (possibly of degree 1). Let Nbe the normal closure of G in G. Then (Theorem 1) either N factorisesas N=GG for some , , or all unfaithful G-orbits, if any exist,are infinite. This result generalises a theorem of I. M. Isaacswhich deals with the case where there is a finite upper boundon the lengths of the G-orbits. Several further results areproved about the structure of G as a permutation group, focussingin particular on the nature of certain G-invariant partitionsof . 1991 Mathematics Subject Classification 20B07, 20B05. 相似文献
6.
This paper considers the finite-element approximation of theelliptic interface problem: -?(u) + cu = f in Rn (n = 2 or3), with u = 0 on , where is discontinuous across a smoothsurface in the interior of . First we show that, if the meshis isoparametrically fitted to using simplicial elements ofdegree k - 1, with k 2, then the standard Galerkin method achievesthe optimal rate of convergence in the H1 and L2 norms overthe approximations l4 of l where l 2. Second, since itmay be computationally inconvenient to fit the mesh to , weanalyse a fully practical piecewise linear approximation ofa related penalized problem, as introduced by Babuska (1970),based on a mesh that is independent of . We show that, by choosingthe penalty parameter appropriately, this approximation convergesto u at the optimal rate in the H1 norm over l4 and in the L2norm over any interior domain l* satisfying l* l** l4 for somedomain l**.
Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton BN1 9QH 相似文献
7.
A Torsion-Free Milnor-Moore Theorem 总被引:1,自引:0,他引:1
Let X be the space of Moore loops on a finite, q-connected,n-dimensional CW complex X, and let R Q be a subring containing1/2. Let (R) be the least non-invertible prime in R. For a gradedR-module M of finite type, let FM = M/Torsion M. We show thatthe inclusion P FH*(X;R) of the sub-Lie algebra of primitiveelements induces an isomorphism of Hopf algebras
provided that (R) n/q. Furthermore, the Hurewiczhomomorphism induces an embedding of F(*(X) R) in P, with P/F(*(X)R)torsion. As a corollary, if X is elliptic, then FH*(X;R) isa finitely generated R-algebra. 相似文献
8.
In this paper, the behaviour of the positive eigenfunction of in u| = 0, p > 1, isstudied near its critical points. Under some convexity and symmetryassumptions on , is seen to have a unique critical point atx = 0; also, the behaviour of both and is determined nearby.Positive solutions u to some general problems pu = f(u)in , u| = 0, are also considered, with some convexity restrictionson u. 2000 Mathematics Subject Classification 35B05 (primary),35J65, 35J70 (secondary). 相似文献
9.
A bifurcation problem governed by the boundary condition II 总被引:1,自引:0,他引:1
Garcia-Melian Jorge; Rossi Julio D.; Sabina De Lis Jose C. 《Proceedings London Mathematical Society》2007,94(1):1-25
In this work we consider the problem u = a(x)up in on , where is a smooth bounded domain, isthe outward unit normal to , is regarded as a parameter and0 < p < 1. We consider both cases where a(x) > 0 in or a(x) is allowed to vanish in a whole subdomain 0 of . Ourmain results include existence of non-negative non-trivial solutionsin the range 0 < < 1, where 1 is characterized by meansof an eigenvalue problem, uniqueness and bifurcation from infinityof such solutions for small , and the appearance of dead coresfor large enough . 相似文献
10.
Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C*-algebraC*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w*-strongly exposed point of P(G)0, where P(G)0={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w*-strongly exposed point of P(G)0, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w*-strongly exposed point of P(G)0. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w*-stronglyexposed point of P(G)0, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the RadonNikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F2 on two generators), there isno closed point in its reduced dual space . 相似文献
11.
The close relationship between the notions of positive formsand representations for a C*-algebra A is one of the most basicfacts in the subject. In particular the weak containment ofrepresentations is well understood in terms of positive forms:given a representation of A in a Hilbert space H and a positiveform on A, its associated representation is weakly containedin (that is, ker ker ) if and only if belongs to the weak*closure of the cone of all finite sums of coefficients of .Among the results on the subject, let us recall the followingones. Suppose that A is concretely represented in H. Then everypositive form on A is the weak* limit of forms of the typex ki=1 i, xi with the i in H; moreover if A is a von Neumannsubalgebra of (H) and is normal, there exists a sequence (i)i 1 in H such that (x) = i 1 i, xi for all x. 相似文献
12.
Let E(Z) = {einx}nZ denote the trigonometrical exponential system.It is well known that E(Z) forms an orthogonal basis in thespace L2(0, 2). In 1964, H. Landau discovered that the trigonometricalsystem has the following property: certain small perturbationsof E(Z) yield exponential systems which are complete in L2 onany finite union of 2-periodic translations of any interval(, 2), 0 < < . 相似文献
13.
The asymptotic expansions of the trace of the heat kernel (t)= [sum ]j=1 exp(-tj) for small positive t, where {j} j=1 arethe eigenvalues of the negative Laplacian -n = -[sum ]nk=1 (/xk)2in Rn (n = 2 or 3), are studied for a general multiply connectedbounded domain which is surrounded by simply connected boundeddomains i with smooth boundaries i (i = 1,...,m), where smoothfunctions Yi (i = 1,...,m) are assuming the Robin boundary conditions(ni + Yi) = 0 on i. Here /ni denote differentiations along theinward-pointing normals to i (i = 1,...,m). Some applicationsof an ideal gas enclosed in the multiply connected bounded containerwith Neumann or Robin boundary conditions are given. 相似文献
14.
Let Ek(z) be the Eisenstein series with weight k for the modulargroup SL(2, ). We prove that the zeros of Ek(ei) interlace withthe zeros of Ek+12(ei) on /2 < < 2/3. That is, any zeroof Ek(ei) lies between two consecutive zeros of Ek+12(ei) on/2 < < 2/3. 相似文献
15.
We introduce new measures of non-compactness for the embeddingoperator Ep,q():Lp1() Lq() and describe their relations withthe essential norm of Ep, q(), local isoperimetricand isocapacitary constants. An explicit formula for the essentialnorm of Ep, q() is obtained for domains with a power cusp onthe boundary and bounded C1 domains. The Neumann problem fora particular Schrödinger operator is discussed on domainswith a power cusp. 相似文献
16.
Let F be a non-Archimedean local field, with the ring of integersoF. Let G = GLN(F), K = GLN (oF), and be a supercuspidal representationof G. We show that there exists a unique irreducible smoothrepresentation of K, such that the restriction to K of a smoothirreducible representation ' of G contains if and only if 'is isomorphic to ° det, where is an unramified quasicharacterof Fx. Moreover, we show that contains with the multiplicity1. As a corollary we obtain a kind of inertial local Langlandscorrespondence. 2000 Mathematics Subject Classification 22E50. 相似文献
17.
Removable singularities for Hardy spaces Hp() = {f Hol(): |f|p u in for some harmonic u}, 0 < p < are studied. A setE = is a weakly removable singularity for Hp(\E) if Hp(\E) Hol(), and a strongly removable singularity for Hp(\E) if Hp(\E)= Hp(). The two types of singularities coincide for compactE, and weak removability is independent of the domain . The paper looks at differences between weak and strong removability,the domain dependence of strong removability, and when removabilityis preserved under unions. In particular, a domain and a setE that is weakly removable for all Hp, but not strongly removablefor any Hp(\E), 0 < p < , are found. It is easy to show that if E is weakly removable for Hp(\E)and q > p, then E is also weakly removable for Hq(\E). Itis shown that the corresponding implication for strong removabilityholds if and only if q/p is an integer. Finally, the theory of Hardy space capacities is extended, anda comparison is made with the similar situation for weightedBergman spaces. 相似文献
18.
Buff Xavier; Fagella NuRia; Geyer Lukas; Henriksen Christian 《Journal London Mathematical Society》2005,72(3):689-716
For (,a) C* x C, let f,a be the rational map defined by f,a(z)= z2 (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f,a has a fixed Hermanring with rotation number (we consider that (e2i,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e2i ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e2i z(1+z). As a consequence, we show that for a (0,1/3), if fl,a has afixed Herman ring with rotation number and if ma is the modulusof the Herman ring, then, as a0, we have e ma=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e(a/2)(z-1/z). 相似文献
19.
The main part of the paper deals with local existence and globalexistence versus blow-up for solutions of the Laplace equationin bounded domains with a non-linear dynamical boundary condition.More precisely, we study the problem consisting in: (1) theLaplace equation in (0, ) x ; (2) a homogeneous Dirichlet condition(0, ) x 0; (3) the dynamical boundary condition ; (4) the initial condition u(0, x) = u0 (x) on . Here is a regular and bounded domain in Rn, with n 1, and0 and 1 endow a measurable partition of . Moreover, m>1,2 p < r, where r = 2 (n 1) / (n 2) whenn 3, r = when n = 1,2, and u0 H1/2 , u0 = 0 on 0. The final part of the paper deals with a refinement of a globalnon-existence result by Levine, Park and Serrin, which is appliedto the previous problem. 2000 Mathematics Subject Classification35K55 (primary), 35K90, 35K77 (secondary). 相似文献
20.
We are interested in the model plasma problem u = u+in ,u = d on , au+ dx=j where is a bounded domain in with boundary ; here, j isa given positive number, the function u and the positive number are the unknowns of the problem, and d is a real parameter.Using a variant of the implicit function theorem, we can provethe existence of a global solution branch parametrized by d.The method has the advantage that it can be used for analysingthe approximation of the above problem by a finite-element method. 相似文献