共查询到20条相似文献,搜索用时 31 毫秒
1.
Yanyan Li 《纯数学与应用数学通讯》1993,46(3):303-340
In this paper, we study the equation –Δu = K(x)u5 in ?3 and provide a large class of positive functions K(x) for which we obtain infinitely many positive solutions which decay at infinity at the rate of |x|?1. © 1993 John Wiley & Sons, Inc. 相似文献
2.
We consider a Brownian snake (Ws,s0) with underlying process a reflected Brownian motion in a bounded domain D. We construct a continuous additive functional (Ls,s0) of the Brownian snake which counts the time spent by the end points s of the Brownian snake paths on D. The random measure Z=sdLs is supported by D. Then we represent the solution v of u=4u2 in D with weak Neumann boundary condition 0 by using exponential moment of (Z,) under the excursion measure of the Brownian snake. We then derive an integral equation for v. For small it is then possible to describe negative solution of u=4u2 in D with weak Neumann boundary condition . In contrast to the exit measure of the Brownian snake out of D, the measure Z is more regular. In particular we show it is absolutely continuous with respect to the surface measure on D for dimension 2 and 3.Mathematics Subject Classification (2000):60J55, 60J80, 60H30, 60G57, 35C15, 35J65 相似文献
3.
Mixed projection methods for systems of variational inequalities 总被引:2,自引:0,他引:2
Let H be a real Hilbert space. Let be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u
*,v
*]∈D(F) × D(K) such that This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection
method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility
assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.
相似文献
4.
N. V. Krylov 《偏微分方程通讯》2013,38(4):687-710
We establish the existence and uniqueness of solutions of fully nonlinear elliptic second-order equations like H(v, Dv, D 2 v, x) = 0 in smooth domains without requiring H to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of H at points at which |D 2 v| ≤K, where K is any given constant. For large |D 2 v| some kind of relaxed convexity assumption with respect to D 2 v mixed with a VMO condition with respect to x are still imposed. The solutions are sought in Sobolev classes. 相似文献
5.
Jean-Stéphane Dhersin Jean-François Le Gall 《Probability Theory and Related Fields》1997,108(1):103-129
We prove a Wiener-type criterion for super-Brownian motion and the Brownian snake.If F is a Borel subset of ℝ
d
and x ∈ ℝ
d
, we provide a necessary and sufficientcondition for super-Brownian motion started at δ
x
to immediately hit the set F. Equivalently, this condition is necessary and sufficient for the hitting time of F by theBrownian snake with initial point x to be 0. A key ingredient of the proof isan estimate showing that the hitting probability of F is comparable, up to multiplicative constants,to the relevant capacity of F. This estimate, which is of independent interest, refines previous results due to Perkins and Dynkin. An important role is
played by additivefunctionals of the Brownian snake, which are investigated here via the potentialtheory of symmetric Markov
processes. As a direct application of our probabilisticresults, we obtain a necessary and sufficient condition for the existence
in a domain D of a positivesolution of the equation Δ; u = u
2
which explodes at a given point of ∂ D.
Received: 5 January 1996 / In revised form: 30 October 1996 相似文献
6.
Wilfried Meidl 《Designs, Codes and Cryptography》2008,46(1):57-65
We show that the linear complexity of a u2
v
-periodic binary sequence, u odd, can easily be calculated from the linear complexities of certain 2
v
-periodic binary sequences. Since the linear complexity of a 2
v
-periodic binary sequence can efficiently be calculated with the Games-Chan algorithm, this leads to attractive procedures
for the determination of the linear complexity of a u2
v
-periodic binary sequence. Realizations are presented for u = 3, 5, 7, 15.
相似文献
7.
We study the boundary layer effect in the small relaxation limit to the equilibrium scalar conservation laws in one space dimension for the relaxation system proposed in [6]. First, it is shown that for initial and boundary data satisfying a strict version of the subcharacteristic condition, there exists a unique global (in time) solution, (uε, vε), to the relaxation system (1.4) for each ε > 0. The spatial total variation of (uε, vε) is shown to be bounded independently of ε, and consequently, a subsequence of (uε, vε) converges to a limit (u, v) as ε → 0+. Furthermore, u(x, t) is a weak solution to the scalar conservation law (1.5) and v = f(u). Next, we prove that for data that are suitably small perturbations of a nontransonic state, the relaxation limit function satisfies the boundary-entropy condition (2.11). Finally, the weak solutions to (1.5) with the boundary-entropy condition (2.11) is shown to be unique. Consequently, the relaxation limit of solutions to (1.4) is unique, and the whole sequence converges to the unique limit. One consequence of our analysis shows that the boundary layer occurs only in the u-component in the sense that vε(0, ·) converges strongly to γ ○ v = f(γ ○ u), the trace of f(u) on the t-axis. © 1998 John Wiley & Sons, Inc. 相似文献
8.
Local-edge-connectivity in digraphs and oriented graphs 总被引:2,自引:0,他引:2
Lutz Volkmann 《Discrete Mathematics》2007,307(24):3207-3212
A digraph without any cycle of length two is called an oriented graph. The local-edge-connectivityλ(u,v) of two vertices u and v in a digraph or graph D is the maximum number of edge-disjoint u-v paths in D, and the edge-connectivity of D is defined as . Clearly, λ(u,v)?min{d+(u),d-(v)} for all pairs u and v of vertices in D. Let δ(D) be the minimum degree of D. We call a graph or digraph D maximally edge-connected when λ(D)=δ(D) and maximally local-edge-connected when
λ(u,v)=min{d+(u),d-(v)} 相似文献
9.
We take up in this paper the existence of positive continuous solutions for some nonlinear boundary value problems with fractional
differential equation based on the fractional Laplacian
(-D|D)\fraca2{(-\Delta _{|D})^{\frac{\alpha }{2}}} associated to the subordinate killed Brownian motion process ZaD{Z_{\alpha }^{D}} in a bounded C
1,1 domain D. Our arguments are based on potential theory tools on ZaD{Z_{\alpha }^{D}} and properties of an appropriate Kato class of functions K
α
(D). 相似文献
10.
Zhi-jian QIU Department of Economic Mathematics Southwestern University of Finance Economics Chengdu China 《中国科学A辑(英文版)》2007,50(3):305-312
For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk. 相似文献
11.
We consider the exit measure of super Brownian motion with a stable branching mechanism of a smooth domain D of ℝ
d
. We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical
dimension. This completes results given by Sheu [22] and generalizes the results of Abraham and Le Gall [2]. Because of the
links between exits measure and partial differential equations, those results imply bounds on solutions of elliptic semi-linear
PDE. We also give the Hausdorff dimension of the support of the exit measure and show it is totally disconnected in high dimension.
Eventually we prove the exit measure is singular with respect to the surface measure on ∂D in the critical dimension. Our main tool is the subordinated Brownian snake introduced by Bertoin, Le Gall and Le Jan [4].
Received: 6 December 1999 / Revised version: 9 June 2000 / Published online: 11 December 2001 相似文献
12.
Jean-François Le Gall 《Probability Theory and Related Fields》1995,102(3):393-432
Summary We investigate the connections between the path-valued process called the Brownian snake and nonnegative solutions of the partial differential equation u=u
2 in a domain of
d
. In particular, we prove two conjectures recently formulated by Dynkin. The first one gives a complete characterization of the boundary polar sets, which correspond to boundary removable singularities for the equation u=u
2. The second one establishes a one-to-one correspondence between nonnegative solutions that are bounded above by a harmonic function, and finite measures on the boundary that do not charge polar sets. This correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the Brownian snake. Our proofs combine probabilistic and analytic arguments. An important role is played by a new version of the special Markov property, which is of independent interest. 相似文献
13.
LetD be a domain inR
2 with the Green functionG(x, y) for the Laplace equation. We give a generalization of the Cranston-McConnell inequality concerning the integrability of positive
harmonic functions onD. A typical new inequality is {fx137-01} whereu andv
1,..., vnare positive superharmonic functions onD andc
nis a constant depending only onn. The generalized Cranston-McConnell inequality is used for the determination of the Martin boundary of a certain unbounded
domain. 相似文献
14.
Zeng Min Song 《Journal of Graph Theory》1994,18(5):461-468
Let D be an oriented graph of order n ≧ 9 and minimum degree n ? 2. This paper proves that D is pancyclic if for any two vertices u and v, either uv ? A(D), or dD+(u) + dD?(v) ≧ n ? 3. 相似文献
15.
To every second-order elliptic differential operator L and to every number α ϵ (1, 2] there is a corresponding measure-valued Markov process X called the (L, α)-superdiffusion. Suppose that Γ is a closed set in Rd. It is known that the following three statements are equivalent: (α) the range of X does not hit Γ; (β) if u ≥ 0 and Lu = uα in Rd\Γ, then u = 0 (in other words, Γ is a removable singularity for all solutions of equation Lu = uα); (γ) Cap2,α′(Γ) = 0 where 1/α + 1/α′ = 1 and Capγ,q is the so-called Bessel capacity. The equivalence of (β) and (γ) was established by Baras and Pierre in 1984 and the equivalence of (α) and (β) was proved by Dynkin in 1991. In this paper, we consider sets Γ on the boundary ∂D of a bounded domain D and we establish (assuming that ∂D is smooth) the equivalence of the following three properties: (a) the range of X in D does not hit Γ (b) if u ≥ 0 and Lu = uα in D, and if u → 0 as x → α ϵ ∂D\Γ, then u = 0; (c) Cap∂2/α,α′(Γ) = 0 where Cap∂γ-qis the Bessel capacity on ∂D. This implies positive answers to two conjectures posed by Dynkin a few years ago. (The conjectures have already been confirmed for α = 2 and L = Δ in a recent paper of Le Gall.) By using a combination of probabilistic and analytic arguments we not only prove the equivalence of (a)-(c) but also give a new, simplified proof of the equivalence of (α)-(γ). The paper consists of an Introduction (Section 1) and two parts, probabilistic (Sections 2 and 3) and analytic (Sections 4 and 5), that can be read independently. An important probabilistic lemma, stated in the Introduction, is proved in the Appendix. © 1996 John Wiley & Sons, Inc. 相似文献
16.
We prove the boundary Harnack principle for ratios of solutions u/v of non-divergence second order elliptic equations Lu = a
ij
D
ij
u + b
i
D
i
u = 0 in a bounded domain Ω ⊂
\mathbb R {\mathbb R}
n
. We assume that b
i
∈ L
n
(Ω) and Ω is a twisted H?lder domain of order α ∈ (1/2, 1]. Based on this result, we derive the H?lder regularity of u/v for uniform domains. Bibliography: 27 titles. 相似文献
17.
In the paper we study the problem of control by means of a heat source g for a thermoelastic system of equations utt − ρ∇· p (θ, ∇u) − νΔut + DΔ2 u = f, cv(θ, ∇u)θt − κΔθ − ρθ[ p θ (θ, ∇u)·∇ut] − ν∣∇ut∣2 = g, in a two-dimensional domain, where both viscosity ν and rigidity D are positive. Such a system has been considered in our former papers, and existence of solutions as well as uniqueness have been obtained. Here we prove the continuity and differentiability of solutions under somewhat stronger assumptions. An example of a control problem and necessary optimality conditions are presented. The system has an interpretation as a plate reinforced with shape memory alloy (SMA) wire mesh. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd. 相似文献
18.
The geodetic numbers of graphs and digraphs 总被引:1,自引:0,他引:1
Chang-hong LU~ 《中国科学A辑(英文版)》2007,50(8):1163-1172
For every two vertices u and v in a graph G,a u-v geodesic is a shortest path between u and v.Let I(u,v)denote the set of all vertices lying on a u-v geodesic.For a vertex subset S,let I(S) denote the union of all I(u,v)for u,v∈S.The geodetic number g(G)of a graph G is the minimum cardinality of a set S with I(S)=V(G).For a digraph D,there is analogous terminology for the geodetic number g(D).The geodetic spectrum of a graph G,denoted by S(G),is the set of geodetic numbers of all orientations of graph G.The lower geodetic number is g~-(G)=minS(G)and the upper geodetic number is g~ (G)=maxS(G).The main purpose of this paper is to study the relations among g(G),g~-(G)and g~ (G)for connected graphs G.In addition,a sufficient and necessary condition for the equality of g(G)and g(G×K_2)is presented,which improves a result of Chartrand,Harary and Zhang. 相似文献
19.
E. B. Dynkin 《Journal of Functional Analysis》2000,170(2):665
This is an attempt to establish a link between positive solutions of semilinear equations Lu=−ψ(u) and Lv=ψ(v) where L is a second order elliptic differential operator and ψ is a positive function. The equations were investigated separately by a number of authors. We try to link them via positive solutions of a linear equation Lu=0 (we call them L-harmonic functions). Let D be an arbitrary open subset of
d and let
(D),
(D) and
(D) stand for the sets of all positive solutions in D for three equations mentioned above. We establish a 1–1 correspondence between certain subclasses of these classes. Similar results are obtained also for the corresponding parabolic equations. A probabilistic interpretation in terms of a superdiffusion is given in [1]. 相似文献
20.
《Discrete Applied Mathematics》2002,116(1-2):115-126
For vertices u and v in an oriented graph D, the closed interval I[u,v] consists of u and v together with all vertices lying in a u−v geodesic or v−u geodesic in D. For S⊆V(D), I[S] is the union of all closed intervals I[u,v] with u,v∈S. A set S is convex if I[S]=S. The convexity number con(D) is the maximum cardinality of a proper convex set of V(D). The nontrivial connected oriented graphs of order n with convexity number n−1 are characterized. It is shown that there is no connected oriented graph of order at least 4 with convexity number 2 and that every pair k, n of integers with 1⩽k⩽n−1 and k≠2 is realizable as the convexity number and order, respectively, of some connected oriented graph. For a nontrivial connected graph G, the lower orientable convexity number con−(G) is the minimum convexity number among all orientations of G and the upper orientable convexity number con+(G) is the maximum such convexity number. It is shown that con+(G)=n−1 for every graph G of order n⩾2. The lower orientable convexity numbers of some well-known graphs are determined, with special attention given to outerplanar graphs. 相似文献