共查询到20条相似文献,搜索用时 31 毫秒
1.
Pavel Drábek Peter Takáč 《Calculus of Variations and Partial Differential Equations》2007,29(1):31-58
An improved Poincaré inequality and validity of the Palais-Smale condition are investigated for the energy functional on , 1 < p < ∞, where Ω is a bounded domain in , is a spectral (control) parameter, and is a given function, in Ω. Analysis is focused on the case λ = λ1, where −λ1 is the first eigenvalue of the Dirichlet p-Laplacian Δ
p
on , λ1 > 0, and on the “quadratization” of within an arbitrarily small cone in around the axis spanned by , where stands for the first eigenfunction of Δ
p
associated with −λ1. 相似文献
2.
For Au = f with an elliptic differential operator and stochastic data f, the m-point correlation function of the random solution u satisfies a deterministic equation with the m-fold tensor product operator A
(m) of A. Sparse tensor products of hierarchic FE-spaces in are known to allow for approximations to which converge at essentially the rate as in the case m = 1, i.e. for the deterministic problem. They can be realized by wavelet-type FE bases (von Petersdorff and Schwab in Appl
Math 51(2):145–180, 2006; Schwab and Todor in Computing 71:43–63, 2003). If wavelet bases are not available, we show here
how to achieve the fast computation of sparse approximations of for Galerkin discretizations of A by multilevel frames such as BPX or other multilevel preconditioners of any standard FEM approximation for A. Numerical examples illustrate feasibility and scope of the method. 相似文献
3.
Zhaoli Liu Jiabao Su Zhi-Qiang Wang 《Calculus of Variations and Partial Differential Equations》2009,35(4):463-480
In this paper, we study existence of nontrivial solutions to the elliptic equation
and to the elliptic system
where Ω is a bounded domain in with smooth boundary ∂Ω, , f (x, 0) = 0, with m ≥ 2 and . Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, for and , and for and , where I
m
is the m × m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity
on the asymptotic behaviors of the nonlinearity f and .
Z. Liu was supported by NSFC(10825106, 10831005). J. Su was supported by NSFC(10831005), NSFB(1082004), BJJW-Project(KZ200810028013)
and the Doctoral Programme Foundation of NEM of China (20070028004). 相似文献
4.
Andrew Raich 《Mathematische Zeitschrift》2007,256(1):193-220
Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional
kernels. We prove that the kernels are C
∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.
相似文献
5.
Xianling Fan Shao-Gao Deng 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(2):255-271
We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving
the p(x)-Laplacian of the form
where Ω is a bounded smooth domain in , and p(x) > 1 for with and φ ≢ 0 on ∂Ω. Using the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that, there exists λ* > 0 such that the problem has at least two positive solutions if λ = λ*, has at least one positive solution if λ = λ*, and has no positive solution if λ = λ*. To prove the result we establish a special strong comparison principle for the Neumann problems.
The research was supported by the National Natural Science Foundation of China 10371052,10671084). 相似文献
6.
In this paper, we have analyzed a one parameter family of hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems with Dirichlet boundary conditions. These methods depend on the values of the parameter , where θ = + 1 corresponds to the nonsymmetric and θ = −1 corresponds to the symmetric interior penalty methods when and f(u,∇u) = −f, that is, for the Poisson problem. The error estimate in the broken H
1 norm, which is optimal in h (mesh size) and suboptimal in p (degree of approximation) is derived using piecewise polynomials of degree p ≥ 2, when the solution . In the case of linear elliptic problems also, this estimate is optimal in h and suboptimal in p. Further, optimal error estimate in the L
2 norm when θ = −1 is derived. Numerical experiments are presented to illustrate the theoretical results.
Supported by DST-DAAD (PPP-05) project. 相似文献
7.
Menita Carozza Chiara Leone Antonia Passarelli di Napoli Anna Verde 《Calculus of Variations and Partial Differential Equations》2009,35(2):215-238
We prove a C
2,α
partial regularity result for local minimizers of polyconvex variational integrals of the type , where Ω is a bounded open subset of , and is a convex function, with subquadratic growth. 相似文献
8.
We consider the problem
where Ω is a bounded smooth domain in , 1 < p< + ∞ if N = 2, if N ≥ 3 and ε is a parameter. We show that if the mean curvature of ∂Ω is not constant then, for ε small enough, such a problem
has always a nodal solution u
ε with one positive peak and one negative peak on the boundary. Moreover, and converge to and , respectively, as ε goes to zero. Here, H denotes the mean curvature of ∂Ω.
Moreover, if Ω is a ball and , we prove that for ε small enough the problem has nodal solutions with two positive peaks on the boundary and arbitrarily
many negative peaks on the boundary.
The authors are supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non
lineari”. 相似文献
9.
Jun Zhang 《Annals of the Institute of Statistical Mathematics》2007,59(1):161-170
The family of α-connections ∇(α) on a statistical manifold equipped with a pair of conjugate connections and is given as . Here, we develop an expression of curvature R
(α) for ∇(α) in relation to those for . Immediately evident from it is that ∇(α) is equiaffine for any when are dually flat, as previously observed in Takeuchi and Amari (IEEE Transactions on Information Theory 51:1011–1023, 2005). Other related formulae are also developed.
The work was conducted when the author was on sabbatical leave as a visiting research scientist at the Mathematical Neuroscience
Unit, RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan. 相似文献
10.
Juha Lehrbäck 《manuscripta mathematica》2008,127(2):249-273
We establish necessary and sufficient conditions for a domain to admit the (p, β)-Hardy inequality , where d(x) = dist(x, ∂Ω) and . Our necessary conditions show that a certain dichotomy holds, even locally, for the dimension of the complement Ω
c
when Ω admits a Hardy inequality, whereas our sufficient conditions can be applied in numerous situations where at least
a part of the boundary ∂Ω is “thin”, contrary to previously known conditions where ∂Ω or Ω
c
was always assumed to be “thick” in a uniform way. There is also a nice interplay between these different conditions that
we try to point out by giving various examples.
The author was supported in part by the Academy of Finland. 相似文献
11.
Given a connected open set
and a function w ∈LN/p(Ω) if 1 < p < N and w ∈Lr (Ω) for some r ∈(1, ∞) if p ≧ N, with
we prove that the positive principal eigenvalue of the problem
is unique and simple. This improves previous works all of which assumed w in a smaller space than LN/p (Ω) to ensure that Harnack’s inequality holds. Our proof does not rely on Harnack’s inequality, which may fail in our case.
Received: 18 March 2005; revised: 7 April 2005 相似文献
12.
Mabel Cuesta Humberto Ramos Quoirin 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(4):469-491
Let Δ
p
denote the p-Laplacian operator and Ω be a bounded domain in . We consider the eigenvalue problem
for a potential V and a weight function m that may change sign and be unbounded. Therefore the functional to be minimized is indefinite and may be unbounded from below.
The main feature here is the introduction of a value α(V, m) that guarantees the boundedness of the energy over the weighted sphere . We show that the above equation has a principal eigenvalue if and only if either m ≥ 0 and α(V, m) > 0 or m changes sign and α(V, m) ≥ 0. The existence of further eigenvalues is also treated here, mainly a second eigenvalue (to the right) and their dependence
with respect to V and m.
相似文献
13.
Let Λ be an algebraic set and let (n is even) be a polynomial mapping such that for each there is r(λ) > 0 such that the mapping g
λ = g(· , λ) restricted to the sphere S
n
(r) is an immersion for every 0 < r < r (λ), so that the intersection number I(g
λ|S
n
(r)) is defined. Then is an algebraically constructible function.
I. Karolkiewicz and A. Nowel supported by the grant BW/5100-5-0286-7. 相似文献
14.
Konrad Gröger Lutz Recke 《NoDEA : Nonlinear Differential Equations and Applications》2006,13(3):263-285
This paper concerns boundary value problems for quasilinear second order elliptic systems which are, for example, of the type
Here Ω is a Lipschitz domain in
νj are the components of the unit outward normal vector field on ∂Ω, the sets Γβ are open in ∂Ω and their relative boundaries are Lipschitz hypersurfaces in ∂Ω. The coefficient functions are supposed to
be bounded and measurable with respect to the space variable and smooth with respect to the unknown vector function u and to the control parameter λ. It is shown that, under natural conditions, such boundary value problems generate smooth
Fredholm maps between appropriate Sobolev-Campanato spaces, that the weak solutions are H?lder continuous up to the boundary
and that the Implicit Function Theorem and the Newton Iteration Procedure are applicable. 相似文献
15.
P. Quittner W. Reichel 《Calculus of Variations and Partial Differential Equations》2008,32(4):429-452
Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂ν
u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L
∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s|
p
) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that
p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) = s
p
then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of
∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential
equation is of the form h(x, u) with h satisfying suitable growth conditions. 相似文献
16.
Nicola Garofalo 《manuscripta mathematica》2008,126(3):353-373
We prove some new a priori estimates for H
2-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group . Such estimates are global and are geometric in nature as they involve the horizontal mean curvature of ∂Ω. As a consequence of our bounds we show that if has step two, then for any smooth H
2-convex function in vanishing on ∂Ω one has
.
Supported in part by NSF Grant DMS-07010001. 相似文献
17.
Leonid Bogachev 《Journal of Theoretical Probability》2006,19(4):849-873
We are concerned with the limit distribution of l
t
-norms (of order t) of samples of i.i.d. positive random variables, as N→∞, t→∞. The problem was first considered by Schlather [(2001), Ann. Probab. 29, 862–881], but the case where {X
i
} belong to the domain of attraction of Gumbel’s double exponential law (in the sense of extreme value theory) has largely
remained open (even for an exponential distribution). In this paper, it is assumed that the log-tail distribution function
is regularly varying at infinity with index . We proceed from studying the limit distribution of the sums , which is of interest in its own right. A proper growth scale of N relative to t appears to be of the form (). We show that there are two critical points, α1 = 1 and α2 = 2, below which the law of large numbers and the central limit theorem, respectively, break down. For α < 2, under a slightly
stronger condition of normalized regular variation of h, we prove that the limit laws for S
N
(t) are stable, with characteristic exponent and skewness parameter . A complete picture of the limit laws for the norms R
N
(t) = S
N
(t)1/t
is then derived. In particular, our results corroborate a conjecture in Schlather [(2001), Ann. Probab. 29, 862–881] regarding the “endpoints” , α→ 0.
相似文献
18.
Analysis of FETI methods for multiscale PDEs 总被引:2,自引:0,他引:2
In this paper, we study a variant of the finite element tearing and interconnecting (FETI) method which is suitable for elliptic
PDEs with highly heterogeneous (multiscale) coefficients α(x); in particular, coefficients with strong variation within subdomains and/or jumps that are not aligned with the subdomain
interfaces. Using energy minimisation and cut-off arguments we can show rigorously that for an arbitrary (positive) coefficient
function the condition number of the preconditioned FETI system can be bounded by C(α) (1 + log(H/h))2 where H is the subdomain diameter and h is the mesh size, and where the function C(α) depends only on the coefficient variation in the vicinity of subdomain interfaces. In particular, if varies only mildly in a layer Ω
i,η
of width η near the boundary of each of the subdomains Ω
i
, then , independent of the variation of α in the remainder Ω
i
\Ω
i,η
of each subdomain and independent of any jumps of α across subdomain interfaces. The quadratic dependence of C(α) on H/η can be relaxed to a linear dependence under stronger assumptions on the behaviour of α in the interior of the subdomains.
Our theoretical findings are confirmed in numerical tests.
C. Pechstein was supported by the Austrian Science Funds (FWF) under grant F1306. 相似文献
19.
Pigong Han Zhaoxia Liu 《Calculus of Variations and Partial Differential Equations》2007,30(3):315-352
Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem
where and Q(x) is a positive continuous function on . Using Moser iteration, we give an asymptotic characterization of solutions for (*) at the origin. Under some conditions
on Q, μ, we, by means of a variational method, prove that there exists such that for every , problem (*) has a positive solution and a pair of sign-changing solutions. 相似文献
20.
We analyze the convergence rate of a multigrid method for multilevel linear systems whose coefficient matrices are generated
by a real and nonnegative multivariate polynomial f and belong to multilevel matrix algebras like circulant, tau, Hartley, or are of Toeplitz type. In the case of matrix algebra
linear systems, we prove that the convergence rate is independent of the system dimension even in presence of asymptotical
ill-conditioning (this happens iff f takes the zero value). More precisely, if the d-level coefficient matrix has partial dimension n
r
at level r, with , then the size of the system is , , and O(N(n)) operations are required by the considered V-cycle Multigrid in order to compute the solution within a fixed accuracy. Since the total arithmetic cost is asymptotically
equivalent to the one of a matrix-vector product, the proposed method is optimal. Some numerical experiments concerning linear
systems arising in 2D and 3D applications are considered and discussed. 相似文献