共查询到20条相似文献,搜索用时 15 毫秒
1.
This article studies the problem of minimizing ∫ΩF(Du)+G(x,u) over the functions uW1,1(Ω) that assume given boundary values on ∂Ω. The function F and the domain Ω are assumed convex. In considering the same problem with G=0, and in the spirit of the classical Hilbert–Haar theory, Clarke has introduced a new type of hypothesis on the boundary function : the lower (or upper) bounded slope condition. This condition, which is less restrictive than the classical bounded slope condition of Hartman, Nirenberg and Stampacchia, is satisfied if is the restriction to ∂Ω of a convex (or concave) function. We show that for a class of problems in which G(x,u) is locally Lipschitz (but not necessarily convex) in u, the lower bounded slope condition implies the local Lipschitz regularity of solutions. 相似文献
2.
We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces Lp()(Ω) with variable exponent. In particular, we show that the maps uLp()(Ω)→k(t)u*Lp*()(0,measΩ) and uLp()(Ω)→u*Lp*()(0,measΩ) are locally -Hölderian (u* (resp. p*) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents. 相似文献
3.
We consider the problem of finding, from the final data u(x,y,T)=g(x,y), the initial data u(x,y,0) of the temperature function u(x,y,t),(x,y)I=(0,π)×(0,π),t[0,T] satisfying the following systemThe problem is severely ill-posed. In this paper a simple and convenient new regularization method for solving this problem is considered. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively. 相似文献
4.
Let G be a graph. For u,vV(G) with distG(u,v)=2, denote JG(u,v)={wNG(u)∩NG(v)|NG(w)NG(u)NG(v){u,v}}. A graph G is called quasi claw-free if JG(u,v)≠ for any u,vV(G) with distG(u,v)=2. In 1986, Thomassen conjectured that every 4-connected line graph is hamiltonian. In this paper we show that every 4-connected line graph of a quasi claw-free graph is hamiltonian connected. 相似文献
5.
Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings 总被引:1,自引:0,他引:1
Let E be a real reflexive Banach space with uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed and convex subset of E. Let T:K→K be a strictly pseudo-contractive map and let L>0 denote its Lipschitz constant. Assume F(T){xK:Tx=x}≠0/ and let zF(T). Fix δ(0,1) and let δ* be such that δ*δL(0,1). Define , where δn(0,1) and limδn=0. Let {αn} be a real sequence in (0,1) which satisfies the following conditions: . For arbitrary x0,uK, define a sequence {xn}K by xn+1=αnu+(1−αn)Snxn. Then, {xn} converges strongly to a fixed point of T. 相似文献
6.
Consider the equation −ε2Δuε + q(x)uε = f(uε) in , u(∞) < ∞, ε = const > 0. Under what assumptions on q(x) and f(u) can one prove that the solution uε exists and limε→0uε = u(x), where u(x) solves the limiting problem q(x)u = f(u)? These are the questions discussed in the paper. 相似文献
7.
R 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2008,25(4):659-678
For a nonnegative, uniformly convex HC2(R2) with H(0)=0, if uC(Ω), ΩR2, is a viscosity solution of the Aronsson equation (1.7), then uC1(Ω). This generalizes the C1-regularity theorem on infinity harmonic functions in R2 by Savin [O. Savin, C1-regularity for infinity harmonic functions in dimensions two, Arch. Ration. Mech. Anal. 176 (3) (2005) 351–361] to the Aronsson equation. 相似文献
8.
In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u + λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k(Ω) in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞). 相似文献
9.
We consider a Cauchy problem for a semilinear heat equation with p>pS where pS is the Sobolev exponent. If u(x,t)=(T−t)−1/(p−1)φ((T−t)−1/2x) for xRN and t[0,T), where φ is a regular positive solution of(P) then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)−1/(p−1) for all p>1. Let pL be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for pS<p<pL. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p>pL. 相似文献
10.
Let m and n be positive integers with n2 and 1mn−1. We study rearrangement-invariant quasinorms R and D on functions f: (0, 1)→
such that to each bounded domain Ω in
n, with Lebesgue measure |Ω|, there corresponds C=C(|Ω|)>0 for which one has the Sobolev imbedding inequality R(u*(|Ω| t))CD(|mu|* (|Ω| t)), uCm0(Ω), involving the nonincreasing rearrangements of u and a certain mth order gradient of u. When m=1 we deal, in fact, with a closely related imbedding inequality of Talenti, in which D need not be rearrangement-invariant, R(u*(|Ω| t))CD((d/dt) ∫{x
n : |u(x)|>u*(|Ω| t)} |(u)(x)| dx), uC10(Ω). In both cases we are especially interested in when the quasinorms are optimal, in the sense that R cannot be replaced by an essentially larger quasinorm and D cannot be replaced by an essentially smaller one. Our results yield best possible refinements of such (limiting) Sobolev inequalities as those of Trudinger, Strichartz, Hansson, Brézis, and Wainger. 相似文献
11.
Wei Dong 《Journal of Mathematical Analysis and Applications》2004,290(2):469-480
In this paper, we consider positive solutions of the logistic type p-Laplacian equation −Δpu=a(x)|u|p−2u−b(x)|u|q−1u, xRN (N2). We show that under rather general conditions on a(x) and b(x) for large |x|, the behavior of the positive solutions for large |x| can be determined. This is then used to show that there is a unique positive solution. Our results improve the corresponding ones in J. London Math. Soc. (2) 64 (2001) 107–124 and J. Anal. Math., in press. 相似文献
12.
Konstantin Yu. Osipenko 《Journal of Approximation Theory》1999,97(2):384
LetSβ{z
: |Im z|<β}. For 2π-periodic functions which are analytic inSβwithp-integrable boundary values, we construct an optimal method of recovery off′(ξ), ξSβ, using information about the valuesf(x1), mldr;, f(xn), xj[0, 2π). 相似文献
13.
Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity 总被引:1,自引:0,他引:1
We consider the semilinear elliptic equation Δu=h(u) in Ω{0}, where Ω is an open subset of (N2) containing the origin and h is locally Lipschitz continuous on [0,∞), positive in (0,∞). We give a complete classification of isolated singularities of positive solutions when h varies regularly at infinity of index q(1,CN) (that is, limu→∞h(λu)/h(u)=λq, for every λ>0), where CN denotes either N/(N−2) if N3 or ∞ if N=2. Our result extends a well-known theorem of Véron for the case h(u)=uq. 相似文献
14.
We investigate the following modification of the well-known irregularity strength of graphs. Given a total weighting w of a graph G=(V,E) with elements of a set {1,2,…,s}, denote wtG(v)=∑evw(e)+w(v) for each vV. The smallest s for which exists such a weighting with wtG(u)≠wtG(v) whenever u and v are distinct vertices of G is called the total vertex irregularity strength of this graph, and is denoted by . We prove that for each graph of order n and with minimum degree δ>0. 相似文献
15.
Letμbe a Gaussian measure (say, onRn) and letK,LRnbe such thatKis convex,Lis a “layer” (i.e.,L={x: ax, ub} for somea, bRanduRn), and the centers of mass (with respect toμ) ofKandLcoincide. Thenμ(K∩L)μ(K)·μ(L). This is motivated by the well-known “positive correlation conjecture” for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses the estimateΦ(x)> 1−((8/π)1/2/(3x+(x2+8)1/2))e−x2/2,x>−1, for the (standard) Gaussian cumulative distribution function, which is sharper than the classical inequality of Komatsu. 相似文献
16.
17.
For α[1,2) we consider operators of the form and for α(0,1) we consider the same operator but where the f term is omitted. We prove, under appropriate conditions on A(x,h), that any solution u to will be in Cα+β if fCβ. 相似文献
18.
Galerkin methods are used to approximate the singular integral equation with solution φ having weak singularity at the endpoint −1, where a, b≠0 are constants. In this case φ is decomposed as φ(x)=(1−x)α(1+x)βu(x), where β=−α, 0<α<1. Jacobi polynomials are used in the discussions. Under the conditions fHμ[−1,1] and k(t,x)Hμ,μ[−1,1]×[−1,1], 0<μ<1, the error estimate under a weighted L2 norm is O(n−μ). Under the strengthened conditions f″Hμ[−1,1] and , 2α−<μ<1, the error estimate under maximum norm is proved to be O(n2α−−μ+), where , >0 is a small enough constant. 相似文献
19.
Ravi P. Agarwal Donal ORegan Patricia J.Y. Wong 《Mathematical and Computer Modelling》2009,50(7-8):999-1025
We consider the system of Hammerstein integral equations where T>0 is fixed, ρi’s are given functions and the nonlinearities fi(t,x1,x2,…,xn) can be singular at t=0 and xj=0 where j{1,2,,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θiui(t)≥0 for t[0,T] and 1≤i≤n, where θi{1,−1} is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained. 相似文献
20.
A norm N on an algebra A is called quadrative if N(x2) ≤ N(x)2 for all x A, and strongly stable if N(xk) ≤ N(x)k for all x A and all k = 2, 3, 4…. Our main purpose in this note is to show that not all quadrative norms are strongly stable. 相似文献