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1.
Summary We consider a (possibly) vector-valued function u: RN, Rn, minimizing the integral , 2-2/(n*1)<p<2, whereD i u=u/x i or some more general functional retaining the same behaviour, we prove higher integrability for Du: D1 u,..., Dn–1 u Lp/(p-1) and Dnu L2; this result allows us to get existence of second weak derivatives: D(D1 u),...,D(Dn–1u)L2 and D(Dn u) L p.This work has been supported by MURST and GNAFA-CNR.  相似文献   

2.
H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp n=, , ,n .. . , 117 234 . . -   相似文献   

3.
    
《Analysis Mathematica》1976,2(3):203-210
B p, (r) (R n ) l l p . B p, (r) (R n ) «» .  相似文献   

4.
We consider the semilinear eigenvalue problem on N (N 2) (N2) and investigate the question under which conditions on the radially symmetric function q, =0 is a bifurcation point for this equation in H1, In H2 and in Lp for 2p+.  相似文献   

5.
Let Mn denote an n-dimensional Riemannian manifold. Its metric is called -strongly spherical if at every point Q Mn there exists a -dimensional subspace Q TQMn such that the curvature operator of the metric of Mn satisfies R(X, Y) Z = k(< Y, Z > X < X, Z > Y), where k = const > 0, Y Q , X, Z #x2208; TQMn. The number is called the index of sphericity and k the exponent of sphericity. The following theorems are proved in the paper.THEOREM 1. Let the Sasakian metric of T1Mn be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if M2 has constant Gaussian curvature K 1 and k = K2/4; b) = 3 if and only if M2 has constant curvature K = 1 and k = 1/4; c) = 0, otherwise.THEOREM 2. Let the Sasakian metric of T1Mn (n Mn) be -strongly spherical with exponent of sphericity k. If k > 1/3 and k 1, then = 0. Let us denote by (Mn, K) a space of constant curvatureK. THEOREM 3. Let the Sasakian metric of T1(Mn, K) (n 3) be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if K = 1/4; b) = 0, otherwise. In dimension n = 3 Theorem 2 is true for k {1/4, 1}.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 150–159, 1992.  相似文献   

6.
Summary We consider the Cauchy problem for the generalized porous medium equation ut=(u) where u=u(x, t), xRn and t>0, and the initial datum u(x, 0) is assumed to be nonnegative, integrable mid to nave compact support. The nonlinearity (u) is a C1 function defined for uO which grows like a power of u. Our assumptions generalize the porous medium case, (u)=um, m>1, and also include the equation of the Marshak waves. This problem has finite speed of propagation. We estimate the rate of growth of the support of the solution with precise estimates for t 0 and t. Our main result deals with the regularity of the solutions. We show that after a certain time t0 the pressure, defined by v=(u), with (u)=(u)/u and (0)=0, is a Lipschitz-continuous function of x and t and the interface is a Lipschitz-continuous surface in RN+1; the solution u is Hölder continuous for all times t> 0.Both authors partially supported by CAICYT, Project 2805-83. The second author also supported by USA-Spain Joint Research Grant CCB-8402023.  相似文献   

7.
Let R(r, m) be the rth order Reed-Muller code of length 2 m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.  相似文献   

8.
This paper deals with polynomial approximations(x) to the exponential function exp(x) related to numerical procedures for solving initial value problems. Motivated by stability requirements, we present a numerical study of the largest diskD()={z C: |z+|} that is contained in the stability regionS()={z C: |(z)|1}. The radius of this largest disk is denoted byr(), the stability radius. On the basis of our numerical study, several conjectures are made concerningr m,p=sup {r(): m,p}. Here m, p (1pm; p, m integers) is the class of all polynomials(x) with real coefficients and degree m for which(x)=exp(x)+O(x p+1) (forx 0).  相似文献   

9.
Let {T1, ..., TN} be a finite set of linear contraction mappings of a Hilbert space H into itself, and let r be a mapping from the natural numbers N to {1, ..., N}. One can form Sn=Tr(n)...Tr(1) which could be described as a random product of the Ti's. Roughly, the Sn converge strongly in the mean, but additional side conditions are necessary to ensure uniform, strong or weak convergence. We examine contractions with three such conditions. (W): xn1, Txn1 implies (I-T)xn0 weakly, (S): xn1, Txn1 implies (I-T)xn0 strongly, and (K): there exists a constant K>0 such that for all x, (I-T)x2K(x2–Tx2).We have three main results in the event that the Ti's are compact contractions. First, if r assumes each value infinitely often, then Sn converges uniformly to the projection Q on the subspace i= 1 N [x|Tix=x]. Secondly we prove that for such compact contractions, the three conditions (W), (S), and (K) are equivalent. Finally if S=S(T1, ..., TN) denotes the algebraic semigroup generated by the Ti's, then there exists a fixed positive constant K such that each element in S satisfies (K) with that K.  相似文献   

10.
Zusammenfassung Für Randwertaufgaben der Form–u–l 0 ...u–l 0 ...u=f(x, u) mitl 0R,lR,f definiert und stetig auf {a<-x<-b, |u|<} wird eine Existenzaussage gewonnen, fallsf inu linear durch die aufeinanderfolgenden Eigenwerte der zugehörigen linearen Aufgabe beschränkt ist. Zum Beweis betrachtet man die äquivalente Hammersteinsche Integralgleichung mit nichtsymmetrischem Kern. Mit Hilfe des Schauderschen Fixpunktsatzes erhält man für diese Integralgleichung Existenzaussagen, welche Ergebnisse von Dolph verallgemeinern.
Summary This note contains an existence theorem for a two-point boundary value problem of the form–u–l 0 u–l 0 u=f(x, u) wherel 0R,l 0R,f defined and continuous on {a<-xb, |u|<} iff is linear bounded inu by the successive eigenvalues of the corresponding linear problem. To prove this result we consider the equivalent Hammerstein integral equation with non-symmetric kernel. Schauders fixpoint principle supplies existence theorems for integral equations of this type which generalize results of Dolph in some sense.
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