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1.
Inverse Sturm–Liouville problems with eigenparameter-dependentboundary conditions are considered. Theorems analogous to thoseof both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)(–(py')'+qy), , where det = > 0, c 0, det > 0, t 0 and (cs + drautb)2 < 4(crta)(dsub). Denoteby (l; ; ) the eigenvalue problem ly = y with boundary conditionsy(0)cos+y'(0)sin = 0 and (a+b)y(1) = (c+d)(py')(1). Define (; ; ) as above but with l replacedby . Let wn denote the eigenfunctionof (l; ; ) having eigenvalue n and initial conditions wn(0)= sin and pw'n(0) = –cos and let n = –awn(1)+cpw'n(1).Define n and n similarly. As sample results, it is proved that if (l; ; ) and (; ; ) have the same spectrum, and (l;; ) and (; ; ) have the samespectrum or for all n, thenq/r = /.  相似文献   

2.
The paper examines blow-up phenomena for the inequality utLu–|u|q–1utL–||q–1 (*) in the half-space x Rn, n 1, where L is a linear second-order partial differential operatorin divergence form. The paper studies weak solutions of (*) that belong only locallyto the corresponding Sobolev spaces in the half-space x Rn. It also requires no conditionsfor the behavior of solutions of (*) on the hyperplane t = 0. The existence of critical blow-up exponents is obtained forsolutions of (*) as a special case of a comparison principlefor the corresponding solutions of (*). For example, the well-knownFujita result is a consequence of the comparison principle. The approach developed in the paper is directly applicable tothe study of analogous problems involving nonlinear differentialoperators. Its elliptic analogue has been recently developedby the authors.  相似文献   

3.
On the Discreteness and Convergence in n-Dimensional Mobius Groups   总被引:5,自引:0,他引:5  
Throughout this paper, we adopt the same notations as in [1,6, 8] such as the Möbius group M(Rn), the Clifford algebraCn–1, the Clifford matrix group SL(2, n), the Cliffordnorm of ||A||=(|a|2+|b|2+|c|2+|d|2) (1) and the Clifford metric of SL(2, n) or of the Möbius groupM(Rn) d(A1,A2)=||A1A2||(|a1a2|2+|b1b2|2+|c1c2|2+|d1d2|2)(2) where |·| is the norm of a Clifford number and represents fi M(), i = 1,2, and so on. In addition, we adopt some notions in [6, 12]:the elementary group, the uniformly bounded torsion, and soon. For example, the definition of the uniformly bounded torsionis as follows.  相似文献   

4.
This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation ut = (um)– up in Rn x (0,) with (1 – 2/n)+ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of ut = (um) with an additional logarithmicanomalous decay exponent in time as t .  相似文献   

5.
The paper investigates the vectorial Dirichlet problem definedby Sj( u(x))=1,xnO; a.e., j=1,...,n, u(x)=\(x),\,x\in|O. end{cases}Here O is an open bounded subset of Rn with boundary |O, andj(A) (j=1,...,n) denote the singular values of the gradient u(x). The existence of solutions is established under one ofthe following assumptions: : O – Rn is continuous on Oand locally contractive on O, or : |O – Rn is contractiveon |O. This extends a result due to Dacorogna and Marcellini.The approach is based on the Baire category method developedearlier by the authors.  相似文献   

6.
Let 1 < p < , 0 < v < p', let be a bounded domainin Rn, and denote by id the limiting compact embedding of theBesov space (Rn) into the exponentialOrlicz space Lexp(tv)(), mapping a function f onto its restrictionf|. In 1993 Triebel established, among others, two-sided estimatesfor the entropy numbers of id, which are even asymptoticallyoptimal for ‘small’ . The aim of the paper is toimprove the upper bounds in the case of ‘large’, where Triebel's estimates are not yet sharp, thus making afurther step towards the conjectured correct asymptotic behaviour.  相似文献   

7.
Let (s, ) be the Hurwitz zeta function with parameter . Powermean values of the form are studied, where q and h are positive integers. These mean valuescan be written as linear combinations of , where r(s1,...,sr;) is a generalization of Euler–Zagiermultiple zeta sums. The Mellin–Barnes integral formulais used to prove an asymptotic expansion of , with respect to q. Hence a general way of deducingasymptotic expansion formulas for is obtained. In particular, the asymptotic expansion of with respect to q is written down.  相似文献   

8.
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   

9.
Consider the following infinite dimensional stochastic evolutionequation over some Hilbert space H with norm |·|: It is proved that under certain mild assumptions, the strongsolution Xt(x0)VHV*, t 0, is mean square exponentially stableif and only if there exists a Lyapunov functional (·,·):HxR+R1 which satisfies the following conditions: (i)c1|x|2k1e–µ1t(x,t)c2|x|2+k2+k2e–µ2t; (ii) L(x,t)–c3(x,t)+k3e–µ3t, xV, t0; where L is the infinitesimal generator of the Markov processXt and ci, ki, µi, i = 1, 2, 3, are positive constants.As a by-product, the characterization of exponential ultimateboundedness of the strong solution is established as the nulldecay rates (that is, µi = 0) are considered.  相似文献   

10.
It is proved that every solution of the Neumann initial-boundaryproblem converges to some equilibrium, if the system satisfies (i) Fi/uj 0 for all 1 i j n, (ii) F(u * g(s)) h(s) * F(u) wheneveru and 0 s 1, where x *y = (x1y1, ..., xnyn) and g, h : [0, 1] [0, 1]n are continuousfunctions satisfying gi(0) = hi(0) = 0, gi(1) = hi(1) = 1, 0< gi(s); hi(s) < 1 for all s (0, 1) and i = 1, 2, ...,n, and (iii) the solution of the corresponding ordinary differentialequation system is bounded in . We also study the convergence of the solution of the Lotka–Volterrasystem where ri > 0, 0, and aij 0 for i j.  相似文献   

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