共查询到20条相似文献,搜索用时 46 毫秒
1.
Razvan Gabriel Iagar Juan Luis Vázquez 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2009
We consider the Dirichlet problem for the p -Laplacian evolution equation, ut=Δpu, where p>2, posed in an exterior domain in RN, with zero Dirichlet boundary condition and with integrable and nonnegative initial data. We are interested in describing the influence of the holes of the domain on the large time behaviour of the solutions. Such behaviour varies depending on the relative values of N and p . We must distinguish between the behaviour near infinity of space (outer analysis), and near the holes (inner analysis). We obtain that the outer analysis is given in all cases by certain self-similar solutions and the inner analysis is given by quasi-stationary states. Logarithmic corrections to exact self-similarity appear in the critical case N=p, which is mathematically more interesting. In this first paper we treat only the cases N>p and N=p, the case N<p will be considered in a companion work. 相似文献
2.
It is shown that if a sequence of open n-sets Dk increases to an open n-set D then reflected stable processes in Dk converge weakly to the reflected stable process in D for every starting point x in D. The same result holds for censored α-stable processes for every x in D if D and Dk satisfy the uniform Hardy inequality. Using the method in the proof of the above results, we also prove the weak convergence of reflected Brownian motions in unbounded domains. 相似文献
3.
By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x), u>0 in Ω, u|∂Ω=∞, where Ω is a bounded domain with smooth boundary in RN, λ>0, g∈C1[0,∞) is increasing on [0,∞), g(0)=0, g′ is regularly varying at infinity with positive index ρ, the weight b, which is non-trivial and non-negative in Ω, may be vanishing on the boundary, and the inhomogeneous term h is non-negative in Ω and may be singular on the boundary. 相似文献
4.
5.
Rolando Magnanini Shigeru Sakaguchi 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2010
We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C2. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C2-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole RN of some substance whose density is initially a characteristic function of the complement of a domain with bounded C2 boundary, and obtain similar results. These results are also extended to the heat flow in the sphere SN and the hyperbolic space HN. 相似文献
6.
The paper deals with the radially symmetric solutions of ut=Δu+um(x,t)vn(0,t), vt=Δv+up(0,t)vq(x,t), subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u (v) blows up alone if and only if m>p+1 (q>n+1), which means that any blow-up is simultaneous if and only if m≤p+1, q≤n+1. (ii) Any blow-up is u (v) blowing up with v (u) remaining bounded if and only if m>p+1, q≤n+1 (m≤p+1, q>n+1). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1, q>n+1. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model. 相似文献
7.
We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer n there are normalized weakly-null sequences of length ωn without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that ωω is the minimal cardinal κ that could possibly have the property that every weakly null κ-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κ which is smaller than the first ω-Erd?s cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either c0 or ?p, with p≥1. 相似文献
8.
9.
In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X in a bounded κ-fat open set; if u is a positive harmonic function with respect to X in a bounded κ-fat open set D and h is a positive harmonic function in D vanishing on Dc, then the non-tangential limit of u/h exists almost everywhere with respect to the Martin-representing measure of h. 相似文献
10.
Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications
We prove that the Schrödinger equation is approximately controllable in Sobolev spaces Hs, s>0, generically with respect to the potential. We give two applications of this result. First, in the case of one space dimension, combining our result with a local exact controllability property, we get the global exact controllability of the system in higher Sobolev spaces. Then we prove that the Schrödinger equation with a potential which has a random time-dependent amplitude admits at most one stationary measure on the unit sphere S in L2. 相似文献
11.
12.
Daniele Cassani Bernhard Ruf Cristina Tarsi 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2010
We study optimal embeddings for the space of functions whose Laplacian Δu belongs to L1(Ω), where Ω⊂RN is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W2,1(Ω) in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when N=2, we establish a sharp embedding inequality into the Zygmund space Lexp(Ω). On one hand, this result enables us to improve the Brezis–Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem Δu=f(x)∈L1(Ω), u=0 on ∂Ω; on the other hand, it represents a borderline case of D.R. Adams' (1988) [1] generalization of Trudinger–Moser type inequalities to the case of higher-order derivatives. Extensions to dimension N?3 are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions. 相似文献
13.
In this paper, we study the Helmholtz equation in a non-smooth inclusion, i.e., in a doubly connected bounded domain B in R2 with boundary ∂B that consists of two disjoint closed curves Γ and Γ0. The existence and uniqueness of a solution to the Helmholtz equation for mixed boundary conditions on Γ are obtained by using Riesz–Fredholm theory. 相似文献
14.
We show how subintegral extensions of certain local Noetherian domains S can be constructed with specified invariants including reduction number, Hilbert function, multiplicity and local cohomology. The construction behaves analytically like Nagata idealization but rather than a ring extension of S, it produces a subring R of S such that R⊆S is subintegral. 相似文献
15.
16.
We study boundary value problems of the form -Δu=f on Ω and Bu=g on the boundary ∂Ω, with either Dirichlet or Neumann boundary conditions, where Ω is a smooth bounded domain in Rn and the data f,g are distributions . This problem has to be first properly reformulated and, for practical applications, it is of crucial importance to obtain the continuity of the solution u in terms of f and g . For f=0, taking advantage of the fact that u is harmonic on Ω, we provide four formulations of this boundary value problem (one using nontangential limits of harmonic functions, one using Green functions, one using the Dirichlet-to-Neumann map, and a variational one); we show that these four formulations are equivalent. We provide a similar analysis for f≠0 and discuss the roles of f and g, which turn to be somewhat interchangeable in the low regularity case. The weak formulation is more convenient for numerical approximation, whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations. We extend the weak formulation to polygonal domains using weighted Sobolev spaces. We also point out some new phenomena for the “concentrated loads” at the vertices in the polygonal case. 相似文献
17.
We exhibit balance conditions between a Young function A and a Young function B for a Korn type inequality to hold between the LB norm of the gradient of vector-valued functions and the LA norm of its symmetric part. In particular, we extend a standard form of the Korn inequality in Lp, with 1<p<∞, and an Orlicz version involving a Young function A satisfying both the Δ2 and the ∇2 condition. 相似文献
18.
Michel Mandjes Petteri Mannersalo Ilkka Norros Miranda van Uitert 《Stochastic Processes and their Applications》2006
Consider events of the form {Zs≥ζ(s),s∈S}, where Z is a continuous Gaussian process with stationary increments, ζ is a function that belongs to the reproducing kernel Hilbert space R of process Z, and S⊂R is compact. The main problem considered in this paper is identifying the function β∗∈R satisfying β∗(s)≥ζ(s) on S and having minimal R-norm. The smoothness (mean square differentiability) of Z turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=s for s∈[0,1] and Z is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process. 相似文献
19.
In many applications it has been observed that hybrid-Monte Carlo sequences perform better than Monte Carlo and quasi-Monte Carlo sequences, especially in difficult problems. For a mixed s-dimensional sequence m, whose elements are vectors obtained by concatenating d-dimensional vectors from a low-discrepancy sequence q with (s−d)-dimensional random vectors, probabilistic upper bounds for its star discrepancy have been provided. In a paper of G. Ökten, B. Tuffin and V. Burago [G. Ökten, B. Tuffin, V. Burago, J. Complexity 22 (2006), 435–458] it was shown that for arbitrary ε>0 the difference of the star discrepancies of the first N points of m and q is bounded by ε with probability at least 1−2exp(−ε2N/2) for N sufficiently large. The authors did not study how large N actually has to be and if and how this actually depends on the parameters s and ε. In this note we derive a lower bound for N, which significantly depends on s and ε. Furthermore, we provide a probabilistic bound for the difference of the star discrepancies of the first N points of m and q, which holds without any restrictions on N. In this sense it improves on the bound of Ökten, Tuffin and Burago and is more helpful in practice, especially for small sample sizes N. We compare this bound to other known bounds. 相似文献