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1.
The Nemitskii operator in the Hölder spaces C0, () with an open bounded subset of Rn, is studied; necessary and sufficientconditions are given for boundedness, uniform continuity, anduniformly continuous differentiability on bounded sets. 相似文献
2.
Sharp Upper Bound to the First Nonzero Neumann Eigenvalue for Bounded Domains in Spaces of Constant Curvature 总被引:1,自引:0,他引:1
The main result of this paper is that for a domain containedin a hemisphere of the n-dimensional sphere Sn the first nonzeroNeumann eigenvalue µ1() is less than or equal to the firstnonzero Neumann eigenvalue µ1(D) where D is a geodesicball in Sn of the same measure as . Equality occurs if and onlyif is isometric to D. This result generalizes old results ofSzegö and Weinberger which gave the corresponding upperbound for µ1() in the Euclidean case, and a result ofChavel for domains in Sn which restricted to lie in a geodesicball of radius when n = 2and to even smaller geodesic balls for larger n. The techniquesused are analogous to those for our recent proof of the Payne-Pólya-Weinbergerconjecture: rearrangement inequalities and properties of specialfunctions are the key elements. The general approach is a directextension of Weinberger's for domains in Rn. 相似文献
3.
Smith Wayne; Stanoyevitch Alexander; Stegenga David A. 《Journal London Mathematical Society》1994,49(2):309-330
We examine two related problems concerning a planar domain .The first is whether Sobolev functions on can be approximatedby global C functions, and the second is whether approximationcan be done by functions in C() which, together with all derivatives,are bounded on . We find necessary and sufficient conditionsfor certain types of domains, such as starshaped domains, andwe construct several examples which show that the general problemis quite difficult, even in the simply connected case. 相似文献
4.
Let R2 be a bounded Lipschitz domain and let
be a Carathèodory integrand such that F(x,·) is polyconvex for L2-a.e. x . Moreover assume thatF is bounded from below and satisfies the condition as det for L2-a.e. x . The paper describes the effect of domain topologyon the existence and multiplicity of strong local minimizersof the functional
wherethe map u lies in the Sobolev space Wid1,p (, R2) with p 2and satisfies the pointwise condition u(x) >0 for L2-a.e.x . The question is settled by establishing that F[·]admits a set of strong local minimizers on that can be indexed by the group Pn Zn, the directsum of Artin's pure braid group on n strings and n copies ofthe infinite cyclic group. The dependence on the domain topologyis through the number of holes n in and the different mechanismsthat give rise to such local minimizers are fully exploitedby this particular representation. 相似文献
5.
Haller-Dintelmann Robert; Heck Horst; Hieber Matthias 《Journal London Mathematical Society》2006,74(3):717-736
Consider a parabolic NxN-system of order m on n with top-ordercoefficients a VMOL. Let 1 < p, q < and let be a Muckenhouptweight. It is proved that systems of this kind possess a uniquesolution u satisfying
whereAu = ||m a Du and J = [0,). In particular, choosing = 1, therealization of A in Lp(n)N has maximal Lp – Lq regularity. 相似文献
6.
Let G be a permutation group on a set , and let m and k be integerswhere 0<m<k. For a subset of , if the cardinalities ofthe sets g\, for gG, are finite and bounded, then is said tohave bounded movement, and the movement of is defined as move()=maxgG|g\|. If there is a k-element subset such that move()m, it is shown that some G-orbit has length at most (k2m)/(km).When combined with a result of P. M. Neumann, this result hasthe following consequence: if some infinite subset has boundedmovement at most m, then either is a G-invariant subset withat most m points added or removed, or nontrivially meets aG-orbit of length at most m2+m+1. Also, if move ()m for allk-element subsets and if G has no fixed points in , then either||k+m (and in this case all permutation groups on have thisproperty), or ||5m2. These results generalise earlierresults about the separation of finite sets under group actionsby B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann,and groups in which all subsets have bounded movement (by theauthor). 相似文献
7.
Hayashi Nakao; Kaikina Elena I.; Naumkin Pavel I. 《Journal London Mathematical Society》2005,72(3):663-688
The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian
where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u0 L1,a L C, u0(x) 0 in Rn, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L1,a C) and the largetime asymptotics are obtained. 相似文献
8.
Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={1, 2, ..., n} be a set of fundamentalroots of , with + being the set of positive roots of with respectto . For and +, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin + will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected. 相似文献
9.
On a smooth curve a theta-characteristic is a line bundle L,the square of which is the canonical line bundle . The equivalentcondition om(L, ) L generalizes well to singular curves, asapplications show. More precisely, a theta-characteristic isa torsion-free sheaf of rank 1 with om(, ) . If the curvehas non-ADE singularities, then there are infinitely many theta-characteristics.Therefore, theta-characteristics are distinguished by theirlocal type. The main purpose of this article is to compute thenumber of even and odd theta-characteristics (that is withh0(C, ) 0 and h0(C, ) 1 modulo 2, respectively) in terms ofthe geometric genus of the curve and certain discrete invariantsof a fixed local type. 相似文献
10.
For reciprocation with respect to a sphere x2=c in Euclideann-space, there is a unitary analogue: Hermitian reciprocationwith respect to an antisphere u=c. This is now applied, forthe first time, to complex polytopes. When a regular polytope has a palindromic Schläfli symbol,it is self-reciprocal in the sense that its reciprocal ', withrespect to a suitable concentric sphere or antisphere, is congruentto . The present article reveals that and ' usually have togetherthe same vertices as a third polytope + and the same facet-hyperplanesas a fourth polytope (where + and are againregular), so as to form a compound, +[2].When the geometry is real, + is the convex hull of and ', while is their common content or core. For instance,when is a regular p-gon {p}, the compound is
The exceptions are of two kinds. In one, + and are notregular. The actual cases are when is an n-simplex {3, 3, ...,3} with n4 or the real 4-dimensional 24-cell {3, 4, 3}=2{3}2{4}2{3}2or the complex 4-dimensional Witting polytope 3{3}3{3}3{3}3.The other kind of exception arises when the vertices of arethe poles of its own facet-hyperplanes, so that , ', + and all coincide. Then is said to be strongly self-reciprocal. 相似文献
11.
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
12.
A sharp geometric condition is given for a bounded domain tohave the property that for each superharmonic function u ona neighbourhood of , there is a superharmonic function on Rnsuch that = u on . 相似文献
13.
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
14.
Let =(n)n1 be a log concave sequence such that lim infn+n/nc>0for some c>0 and ((log n)/n)n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||Tn||=O(n)as n tends to + with n11/(n log n)=+, then T is unitary. Onthe other hand, if n11/(n log n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||Tn||=O(n) as n tends to +, andlim supn+||Tn||=+. 相似文献
15.
Weighted Hardy-Type Inequalities for Differences and the Extension Problem for Spaces with Generalized Smoothness 总被引:1,自引:0,他引:1
It is well known that there are bounded domains Rn whose boundaries are not smooth enough for there to exist a bounded linear extensionfor the Sobolev space into , but the embedding is nevertheless compact. For the Lipboundaries (0<<1) studied in [3, 4], there does not existin general an extension operator of into but there is a bounded linear extension of into and the smoothness retained by thisextension is enough to ensure that the embedding is compact. It is natural to ask if this is typicalfor bounded domains which are such that is compact, that is, that there exists a boundedextension into a space of functions in Rn which enjoy adequatesmoothness. This is the question which originally motivatedthis paper. Specifically we study the extension by zerooperator on a space of functions with given generalizedsmoothness defined on a domain with an irregular boundary, anddetermine the target space with respect to which it is bounded. 相似文献
16.
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
17.
We study the following question: given an open set , symmetricabout 0, and a continuous, integrable, positive definite functionf, supported in and with f(0) = 1, how large can f be? Thisproblem has been studied so far mostly for convex domains inEuclidean space. In this paper we study the question in arbitrarylocally compact abelian groups and for more general domains.Our emphasis is on finite groups as well as Euclidean spacesand d. We exhibit upper bounds for f assuming geometric propertiesof of two types: (a) packing properties of and (b) spectralproperties of . Several examples and applications of the maintheorems are shown. In particular, we recover and extend severalknown results concerning convex domains in Euclidean space.Also, we investigate the question of estimating f over possiblydispersed sets solely in dependence of the given measure m :=||of . In this respect we show that in and the integral is maximalfor intervals. 相似文献
18.
If is an automorphism and is a -derivation of a ring R, thenthe subring of invariants is the set R()={rR|(r)=0}. The mainresult of this paper is let R be a semiprime ring withan algebraic -derivation such that R() is central; then R iscommutative. This theorem generalizes results on theinvariants of automorphisms and derivations and is proved byreducing down to the special cases of automorphisms and derivations. 相似文献
19.
We study concentration phenomena for the system
in the unit ball B1 of 3 with Dirichlet boundaryconditions. Here , , > 0 and p > 1. We prove the existenceof positive radial solutions (, ) such that concentrates ata distance (/2)|log | away from the boundary B1 as the parameter tends to 0. The approach is based on a combination of LyapunovSchmidtreduction procedure together with a variational method. 相似文献
20.
The Beurling algebras l1(D,)(D=N,Z) that are semi-simple, withcompact Gelfand transform, are considered. The paper gives anecessary and sufficient condition (on ) such that l1(D,) possessesa uniform quantitative version of Wiener's theorem in the sensethat there exists a function :]0,+[]0,+ such that, for everyinvertible element x in the unit ball of l1(D,), we have ||x1||(r(x1)) r(x1) is the spectral radiusof x1. 相似文献