共查询到20条相似文献,搜索用时 93 毫秒
1.
The Cheeger problem for a bounded domain Ω⊂RN, N>1 consists in minimizing the quotients |∂E|/|E| among all smooth subdomains E⊂Ω and the Cheeger constant h(Ω) is the minimum of these quotients. Let be the p-torsion function, that is, the solution of torsional creep problem −Δp?p=1 in Ω, ?p=0 on ∂Ω, where Δpu:=div(|∇u|p−2∇u) is the p-Laplacian operator, p>1. The paper emphasizes the connection between these problems. We prove that . Moreover, we deduce the relation limp→1+‖?p‖L1(Ω)?CNlimp→1+‖?p‖L∞(Ω) where CN is a constant depending only of N and h(Ω), explicitely given in the paper. An eigenfunction u∈BV(Ω)∩L∞(Ω) of the Dirichlet 1-Laplacian is obtained as the strong L1 limit, as p→1+, of a subsequence of the family {?p/‖?p‖L1(Ω)}p>1. Almost all t-level sets Et of u are Cheeger sets and our estimates of u on the Cheeger set |E0| yield |B1|hN(B1)?|E0|hN(Ω), where B1 is the unit ball in RN. For Ω convex we obtain u=|E0|−1χE0. 相似文献
2.
Vitali Liskevich I.I. Skrypnik 《Journal of Mathematical Analysis and Applications》2008,338(1):536-544
We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a0(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|p−2+u|u|q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron. 相似文献
3.
Yufei Zhao 《Journal of Number Theory》2010,130(5):1212-1220
A more sums than differences (MSTD) set is a finite subset S of the integers such that |S+S|>|S−S|. We construct a new dense family of MSTD subsets of {0,1,2,…,n−1}. Our construction gives Θ(n2/n) MSTD sets, improving the previous best construction with Ω(n2/n4) MSTD sets by Miller, Orosz, and Scheinerman. 相似文献
4.
Jean-Luc Paillet 《Journal of Combinatorial Theory, Series B》1979,26(2):150-153
Let F be any family of subsets of a finite set E and let n be an integer, n<|F|. Under what condition does the knowledge of cardinals of m-intersections in F, for all m≤n, univocally determine the cardinal of any intersection in F, and what is the minimal condition? We give a complete answer to that. For any n, this determination property is satisfied by n if and only if |E|<2n, without further condition on F. 相似文献
5.
Let be a continuous function such that H(p)→H0∈R as |p|→+∞. Fixing a domain Ω in R2 we study the behaviour of a sequence (un) of approximate solutions to the H-system Δu=2H(u)ux∧uy in Ω. Assuming that supp∈R3|(H(p)−H0)p|<1, we show that the weak limit of the sequence (un) solves the H-system and un→u strongly in H1 apart from a countable set S made by isolated points. Moreover, if in addition H(p)=H0+o(1/|p|) as |p|→+∞, then in correspondence of each point of S we prove that the sequence (un) blows either an H-bubble or an H0-sphere. 相似文献
6.
This note is devoted to a generalization of the Strassen converse. Let gn:R∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X1,X2,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X1,X2,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result. 相似文献
7.
S. B. Vakarchuk 《Ukrainian Mathematical Journal》1996,48(4):601-606
In the spaces E q(Ω), 1 < q < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral n-widths of the classes W r E p(Ω) and W r E p(Ω)Ф in the case where p and q are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration. 相似文献
8.
Ahmed Hamydy 《Journal of Mathematical Analysis and Applications》2010,371(2):534-169
Assume that Ω is a bounded domain in RN (N?3) with smooth boundary ∂Ω. In this work, we study existence and uniqueness of blow-up solutions for the problem −Δp(u)+c(x)|∇u|p−1+F(x,u)=0 in Ω, where 2?p. Under some conditions related to the function F, we give a sufficient condition for existence and nonexistence of nonnegative blow-up solutions. We study also the uniqueness of these solutions. 相似文献
9.
We give the boundedness on Triebel-Lizorkin spaces for oscillatory singular integral operators with polynomial phases and rough kernels of the form eiP(x)Ω(x)|x|−n, where Ω∈Llog+L(Sn−1) is homogeneous of degree zero and satisfies certain cancellation condition. 相似文献
10.
Ferenc Móricz 《Analysis Mathematica》2010,36(4):275-286
We consider the Walsh orthonormal system on the interval [0, 1) in the Paley enumeration and the Walsh-Fourier coefficients $ \hat f $ (n), n ∈ ?, of functions f ∈ L p for some 1 < p ≤ 2. Our aim is to find best possible sufficient conditions for the finiteness of the series Σ n=1 ∞ a n | $ \hat f $ (n)| r , where {a n } is a given sequence of nonnegative real numbers satisfying a mild assumption and 0 < r < 2. These sufficient conditions are in terms of (either global or local) dyadic moduli of continuity of f. The sufficient conditions presented in the monograph [2] are special cases of our ones. 相似文献
11.
We consider equations (E) −Δu+g(u)=μ in smooth bounded domains Ω⊂RN, where g is a continuous nondecreasing function and μ is a finite measure in Ω. Given a bounded sequence of measures (μk), assume that for each k?1 there exists a solution uk of (E) with datum μk and zero boundary data. We show that if uk→u# in L1(Ω), then u# is a solution of (E) relative to some finite measure μ#. We call μ# the reduced limit of (μk). This reduced limit has the remarkable property that it does not depend on the boundary data, but only on (μk) and on g. For power nonlinearities g(t)=|t|q−1t, ∀t∈R, we show that if (μk) is nonnegative and bounded in W−2,q(Ω), then μ and μ# are absolutely continuous with respect to each other; we then produce an example where μ#≠μ. 相似文献
12.
We study certain hypersingular integrals TΩ,α,βf defined on all test functions f∈S(Rn), where the kernel of the operator TΩ,α,β has a strong singularity |y|−n−α(α>0) at the origin, an oscillating factor ei|y|−β(β>0) and a distribution Ω∈Hr(Sn−1), 0<r<1. We show that TΩ,α,β extends to a bounded linear operator from the Sobolev space to the Lebesgue space Lp for β/(β−α)<p<β/α, if the distribution Ω is in the Hardy space Hr(Sn−1) with 0<r=(n−1)/(n−1+γ)(0<γ?α) and β>2α>0. 相似文献
13.
We study the boundary value problems for Monge-Ampère equations: detD2u=e−u in Ω⊂Rn, n?1, u|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD2u=e−tu in Ω, u|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure. 相似文献
14.
In this paper, we study the solvability of the Steklov problem Δpu=|u|p−2u in Ω, on ∂Ω, under assumptions on the asymptotic behaviour of the quotients f(x,s)/|s|p−2s and pF(x,s)/|s|p which extends the classical results with Dirichlet boundary conditions that for a.e. x∈∂Ω, the limits at the infinity of these quotients lie between the first two eigenvalues. 相似文献
15.
Khalid Koufany 《Journal of Functional Analysis》2006,236(2):546-580
Let Ω be a bounded symmetric domain of non-tube type in Cn with rank r and S its Shilov boundary. We consider the Poisson transform Psf(z) for a hyperfunction f on S defined by the Poisson kernel Ps(z,u)=s(h(z,z)n/r/2|h(z,u)n/r|), (z,u)×Ω×S, s∈C. For all s satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in Mn,m(C) (n?m), we prove that an eigenvalue equation for the second order Mn,n-valued Hua operator characterizes the image. 相似文献
16.
Younggi Choi 《Journal of Pure and Applied Algebra》2011,215(9):2119-2129
We study the mod 2 homology of the double and triple loop spaces of homogeneous spaces associated with exceptional Lie groups. The main computational tools are the Serre spectral sequence for fibrations Ωn+1G→Ωn+1(G/H)→ΩnH for n=1,2, and the Eilenberg-Moore spectral sequence associated with related fiber squares, which both converge to the same destination space H∗(Ωn(G/H);F2). We also develop the generalized Bockstein lemma to determine the higher Bockstein actions. 相似文献
17.
ZhangJie Wang 《中国科学 数学(英文版)》2016,59(11):2145-2166
We study 2-primary parts ⅢX(E~((n))/Q)[2~∞] of Shafarevich-Tate groups of congruent elliptic curves E~((n)): y~2= x~3-n~2x, n ∈Q~×/Q~(×2). Previous results focused on finding sufficient conditions for ⅢX(E~((n))/Q)[2~∞]trivial or isomorphic to(Z/2Z)~2. Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E~((n))is isomorphic to(Z/2Z)~2 and the Mordell-Weil rank of E~((n)) is zero,provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for ⅢX(E~((n))/Q)[2~∞]■(Z/2Z)~(2k), where k≥2. 相似文献
18.
Mihai Mih?ilescu 《Journal of Mathematical Analysis and Applications》2007,330(1):416-432
We study the boundary value problem −div(log(1+q|∇u|)|∇u|p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary. We distinguish the cases where either f(u)=−λ|u|p−2u+|u|r−2u or f(u)=λ|u|p−2u−|u|r−2u, with p, q>1, p+q<min{N,r}, and r<(Np−N+p)/(N−p). In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces. 相似文献
19.
David Kalaj 《Journal of Mathematical Analysis and Applications》2011,379(2):783-787
In this note we determine all numbers q∈R such that q|u| is a subharmonic function, provided that u is a K-quasiregular harmonic mappings in an open subset Ω of the Euclidean space Rn. 相似文献
20.
Removable singularity of the polyharmonic equation 总被引:1,自引:0,他引:1
Shu-Yu Hsu 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(2):624-627
Let x0∈Ω⊂Rn, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δmu=0 in Ω?{x0} has a removable singularity at x0 if and only if as |x−x0|→0 for n≥3 and as |x−x0|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x0 if |u(x)|=o(|x−x0|2m−n) as |x−x0|→0 for n≥3 and |u(x)|=o(|x−x0|2m−2log(|x−x0|−1)) as |x−x0|→0 for n=2. 相似文献