We show that in a smooth bounded domain , n⩾2, all global nonnegative solutions of ut−Δum=up with zero boundary data are uniformly bounded in by a constant depending on and τ but not on u0, provided that 1<m<p<[(n+1)/(n−1)]m. Furthermore, we prove an a priori bound in depending on under the optimal condition 1<m<p<[(n+2)/(n−2)]m. 相似文献
In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation ut=div(um−1|∇u|p−2∇u)+Vum+p−2 in a cylinder , with initial condition u(·,0)=u0(·)⩾0 and vanishing on the parabolic boundary . Here (resp. ) is a bounded domain with smooth boundary, , , 1<p<N and m+p−2>0. The critical exponents are found and the nonexistence results are proved for . 相似文献
We study positive solutions of the equation ?ε2Δu+u=up, where p>1 and ε>0 is small, with Neumann boundary conditions in a three-dimensional domain . We prove the existence of solutions concentrating along some closed curve on . To cite this article: A. Malchiodi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).相似文献
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
Variational problems for the multiple integral , where and are studied. A new condition on g, called W1,p-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of in and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in , p ? n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses. 相似文献
Recent articles by Kushner and Meisner (1980) and Kushner, Lebow and Meisner (1981) have posed the problem of characterising the ‘EP’ functions f(S) for which Ef(S) for which E(f(S)) = λnf(Σ) for some λn ? , whenever the m × m matrix S has the Wishart distribution W(m, n, Σ). In this article we obtain integral representations for all nonnegative EP functions. It is also shown that any bounded EP function is harmonic, and that EP polynomials may be used to approximate the functions in certain Lp spaces. 相似文献
Let Wm,p denote the Sobolev space of functions on n whose distributional derivatives of order up to m lie in Lp(n) for 1 ? p ? ∞. When 1 < p < ∞, it is known that the multipliers on Wm,p are the same as those on Lp. This result is true for p = 1 only if n = 1. For, we prove that the integrable distributions of order ?1 whose first order derivatives are also integrable of order ?1, belong to the class of multipliers on Wm,1 and there are such distributions which are not bounded measures. These distributions are also multipliers on Lp, for 1 < p < ∞. Moreover, they form exactly the multiplier space of a certain Segal algebra. We have also proved that the multipliers on Wm,l are necessarily integrable distributions of order ?1 or ?2 accordingly as m is odd or even. We have obtained the multipliers from L1(n) into Wm,p, 1 ? p ? ∞, and the multiplier space of Wm,1 is realised as a dual space of certain continuous functions on n which vanish at infinity. 相似文献
Let [(Ω)]p be the Cartesian product of the space of real-valued infinitely differentiable functions on a connected open set Ω in n with itself p-times. The finitely generated submodules of [(Ω)]p are of the form im(F) where F: [(Ω)]q → [(Ω)]p is a p × q matrix of infinitely differentiable functions on Ω. Let . The main results of the present paper are that for Ω ? n, if the finitely generated submodule im(F) is closed in [(Ω)]p, then for every x?ω with rank(F(x)) < r there exists an r × r sub-matrix A of F such that x is a zero of finite order of det(A), and for Ω ? 1 the converse also holds. 相似文献
Let be a domain with Lipschitzian boundary of a compact Riemannian manifold (M,g) and p>1. We prove that we can make the volume of M arbitrarily close to the volume of while the first eigenvalue of the p-Laplacian on M remains uniformly bounded from below in terms of the the first eigenvalue of the Neumann problem for the p-Laplacian on . To cite this article: A.-M. Matei, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 255–258.相似文献
Let A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (?2(m ? n)A + B + λ2nI)u? = f and (B + λ2nI)u = f in a bounded domain Ω in Rk with a smooth boundary ?Ω. The estimate is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ?, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given. 相似文献
We consider, for odd primes p, the function N(p, m, α) which equals the number of subsets S?{1,…,p ? 1} with the property that Σ∞∈Sxm ≡ α (mod p). We obtain a closed form expression for N(p, m, α). We give simple explicit formulas for N(p, 2, α) (which in some cases involve class numbers and fundamental units), and show that for a fixed m, the difference between N(p, m, α) and its average value p?12p?1 is of the order of or less. Finally, we obtain the curious result that if p ? 1 does not divide m, then N(p, m, 0) > N(p, m, α) for all α ? 0 (mod p). 相似文献
Let Ω ? N be an open set with and some l > 0 satisfying an additional regularity condition. We give asymptotic estimates for the approximation numbers αn of Sobolev imbeddings over these quasibounded domains Ω. Here denotes the Sobolev space obtained by completing under the usual Sobolev norm. We prove , where . There are quasibounded domains of this type where γ is the exact order of decay, in the case p ? q under the additional assumption that either 1 ? p ? q ? 2 or 2 ? p ? q ? ∞. This generalizes the known results for bounded domains which correspond to l = ∞. Similar results are indicated for the Kolmogorov and Gelfand numbers of Ip,qm. As an application we give the rate of growth of the eigenvalues of certain elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded domain of the above type. 相似文献
Asymptotic expansions are given for the distributions of latent roots of matrices in three multivariate situations. The distribution of the roots of the matrix S1(S1 + S2)?1, where S1 is and S2 is Wm(n2, Σ), is studied in detail and asymptotic series for the distribution are obtained which are valid for some or all of the roots of the noncentrality matrix Ω large. These expansions are obtained using partial-differential equations satisfied by the distribution. Asymptotic series are also obtained for the distributions of the roots of n?1S, where S in Wm(n, Σ), for large n, and S1S2?1, where S1 is Wm(n1, Σ) and S2 is Wm(n2, Σ), for large n1 + n2. 相似文献
This article discusses linear differential boundary systems, which include nth-order differential boundary relations as a special case, in np[0,1] × np[0,1], 1 ? p < ∞. The adjoint relation in nq[0,1] × nq[0,1], , is derived. Green's formula is also found. Self-adjoint relations are found in n2[0,1] × n2[0,1], and their connection with Coddington's extensions of symmetric operators on subspaces of np[0,1] × n2[0,1] is established. 相似文献
Let be a Dirichlet form in , where Ω is an open subset of n, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let k be a Dirichlet form on some k-dimensional submanifold of Ω. The paper is devoted to the study of the closability of the forms E with domain and defined by: ki where 1 ? kp < ? < n, and where , gki denote restrictions of ?, g in to . Conditions are given for E to be closable if, for each i = 1,…, p, one has ki = n ? i. Other conditions are given for E to be nonclosable if, for some i, ki < n ? i. 相似文献
Theorem. There is a countable admissible set, , with ordinal ωCK1 such that if S is Sacks generic over then ω1S > ωCK1 and S is a nonminimal upper bound for the hyperdegrees in . (The same holds over for any upper bound produced by any forcing which can be construed so that the forcing relation for Σ1 formulas is Σ1.) A notion of forcing, the “delayed collapse” of ωCK1, is defined. The construction hinges upon the symmetries inherent in how this forcing interacts with Σ1 formulas. It also uses Steel trees to make a certain part of the generic object Σ1 over the final inner model, , and, indeed, over many generic extensions of . 相似文献
While it has already been demonstrated that the set of twin primes (primes that differ by 2) is scarce in the (all twin primes) converges whereas (all primes) diverges, this paper proves in Theorems 1 and 2 the scarcity of twin primes (and, in general, of primes p which differ by any even integer as well as primes p for which yp + z is prime, y positive, z nonzero, (y, z) = 1) in a novel and natural way — by showing that the natural density of such primes compared to the set of all primes is 0, that is, , where π′(n) is the number of, say, twin primes between 1 and n for any n, and π(n) is the number of all primes between 1 and n. Theorem 3 then establishes that if a set of primes is scarce in the sense that the sum of the reciprocals of such primes converges, they are also scarce in the natural density sense outlined above. 相似文献
We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear 2mth order parabolic equation ut=?(?Δ)mu+|u|p in RN×R+, where m>1, p>1, with bounded integrable initial data u0. We prove that in the supercritical Fujita range p>pF=1+2m/N any small global solution with nonnegative initial mass, , exhibits as t→∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case where solutions can blow-up for any arbitrarily small initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents , where p0=pF, are discussed. To cite this article: Yu.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 805–810.相似文献
For a > 0 let , the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about () that , where pa(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of pa(u) is worked out. 相似文献