Let (m, n) ∈ ℕ2, Ω an open bounded domain in ℝm, Y = [0, 1]m; uε in (L2(Ω))n which is two-scale converges to some u in (L2(Ω × Y))n. Let φ: Ω × ℝm × ℝn → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝm × ℝn, φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C1 > 0 and a function C2 ∈ L2(Ω) such that
Let U = ℂ2, Γ = ℤ2, and let ℂ[x1±1, x2±1] be the ring of Laurent polynomials. The Witt algebra L is the Lie algebra of derivations over ℂ[x1±1, x2±1], which is spanned by elements of the form D(u, r) = xr(u1d1 + u2d2), u = (u1, u2) ∈ U, r ∈ Γ, where d1 and d2 are the degree derivations of ℂ[x1±1, x2±1]. The image of gl2-module V under Larsson functor Fα, denoted by W = Fα(V), gives a class of L-modules, often called the Larsson-modules of L. In this paper, we study the derivations from the Witt algebra L to its Larsson-modules W, and we determine the first cohomology group H1(L,W). 相似文献
We improve some previous existence and nonexistence results for positive principal eigenvalues of the problem —Δpu = λg(x)ψp(u), x ∈ ℝN, lim‖x‖⇒+∞u(x) = 0. Also we discuss existence, nonexistence and antimaximum principle questions concerning the perturbed problem —Δpu = λg(x)ψp(u) + f(x), x∈ ℝN. 相似文献
The aim of this note is to generalize a result of Barron [1] concerning the approximation of functions, which can be expressed in terms of the Fourier transform, by superpositions of a fixed sigmoidal function. In particular, we consider functions of the type h(x) = ∫ℝd ƒ (〈t, x〉)dμ(t), where μ is a finite Radon measure on ℝd and ƒ : ℝ → ℂ is a continuous function with bounded variation in ℝ We show (Theorem 2.6) that these functions can be approximated in L2-norm by elements of the set Gn = {Σi=0staggeredn cig(〈ai, x〉 + bi) : aiℝd, bi, ciℝ}, where g is a fixed sigmoidal function, with the error estimated by C/n1/2, where C is a positive constant depending only on f. The same result holds true (Theorem 2.9) for f : ℝ → ℂ satisfying the Lipschitz condition under an additional assumption that ∫ℝd6t6ed|u(t)| > ∞ 相似文献
Let W ì \Bbb Rn\Omega \subset {\Bbb R}^n be a smooth domain and let u ? C0(W).u \in C^0(\Omega ). A classical result of potential theory states that¶¶-òSr([`(x)])u(x)ds(x)=u([`(x)])-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x)=u(\bar x)¶¶for every [`(x)] ? W\bar x\in \Omega and r > 0r>0 if and only if¶¶Du=0 in W.\Delta u=0 \hbox { in } \Omega.¶¶Here -òSr([`(x)])u(x)ds(x)-\kern-5mm\int\limits _{S_{r}(\bar x)} u(x)d\sigma (x) denotes the average of u on the sphere Sr([`(x)])S_r(\bar x) of center [`(x)]\bar x and radius r. Our main result, which is a "localized" version of the above result, states:¶¶Theorem. Let u ? W2,1(W)u\in W^{2,1}(\Omega ) and let x ? Wx\in \Omega be a Lebesgue point of Du\Delta u such that¶¶-òSr([`(x)])ud s- a = o(r2)-\kern-5mm\int\limits _{S_{r}(\bar x)} u d \sigma - \alpha =o(r^2)¶¶for some a ? \Bbb R\alpha \in \Bbb R and all sufficiently small r > 0.r>0. Then¶¶Du(x)=0.\Delta u(x)=0. 相似文献
We consider two quasi-linear initial-value Cauchy problems on ?d: a parabolic system and an hyperbolic one. They both have a first order non-linearity of the form φ(t, x, u)·?u, a forcing term h(t, x, u) and an initial condition u0 ∈ L∞(?d) ∩ C∞(?d), where φ (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t, x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but a direct construction based on parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method. 相似文献
We study convergence properties of {υ(∇uk)}k∈ℕ if υ ∈ C(ℝm×m), |υ(s)| ⩽ C(1+|s|p), 1 < p < + ∞, has a finite quasiconvex envelope, uk → u weakly in W1,p (Ω; ℝm) and for some g ∈ C(Ω) it holds that ∫Ωg(x)υ(∇uk(x))dx → ∫Ωg(x)Qυ(∇u(x))dx as k → ∞. In particular, we give necessary and sufficient conditions for L1-weak convergence of {det ∇uk}k∈ℕ to det ∇u if m = n = p.
Dedicated to Jiří V. Outrata on the occasion of his 60th birthday
This work was supported by the grants IAA 1075402 (GA AV ČR) and VZ6840770021 (MŠMT ČR). 相似文献
Let k(y) > 0, 𝓁(y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and limy → 0k(y)/𝓁(y) exists; then the equation L(u) ≔ k(y)uxx – ∂y(𝓁(y)uy) + a(x, y)ux = f(x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f(x, y, u) in G, u|AC = 0, where G is a simply connected domain in ℝ2 with piecewise smooth boundary ∂G = AB∪AC∪BC; AB = {(x, 0) : 0 ≤ x ≤ 1}, AC : x = F(y) = ∫y0(k(t)/𝓁(t))1/2dt and BC : x = 1 – F(y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f(x, y, u) satisfies Carathéodory condition and |f(x, y, u)| ≤ Q(x, y) + b|u| with Q ∈ L2(G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption |f(x, y, u1) – f(x, y, u2| ≤ C|u1 – u2|, where C = const > 0. 相似文献
Let G be a 2-connected graph with maximum degree Δ (G)≥d, and let x and y be distinct vertices of G. Let W be a subset of V(G)−{x, y} with cardinality at most d−1. Suppose that max{dG(u), dG(v)}≥d for every pair of vertices u and v in V(G)−({x, y}∪W) with dG(u,v)=2. Then x and y are connected by a path of length at least d−|W|.
Received: February 5, 1998 Revised: April 13, 1998 相似文献
Let R be a prime ring of char R ≠ 2, d a nonzero derivation of R, U a noncentral Lie ideal of R, and a ∈ R. If aun1d(u)n2un3d(u)n4un5… d(u)nk?1unk = 0 for all u ∈ U, where n1, n2,…,nk are fixed non-negative integers not all zero, then a = 0 and if a(usd(u)ut)n ∈ Z(R) for all u ∈ U, where s ≥ 0, t ≥ 0, n ≥ 1 are some fixed integers, then either a = 0 or R satisfies S4, the standard identity in four variables. 相似文献
Let t = (t1,…,tn) be a point of ?n. We shall write . We put, by the definition, Wα(u, m) = (m?2u)(α ? n)/4[π(n ? 2)/22(α + n ? 2)/2Г(α/2)]J(α ? n)/2(m2u)1/2; here α is a complex parameter, m a real nonnegative number, and n the dimension of the space. Wα(u, m), which is an ordinary function if Re α ≥ n, is an entire distributional function of α. First we evaluate {□ + m2}Wα + 2(u, m) = Wα(u, m), where {□ + m2} is the ultrahyperbolic operator. Then we express Wα(u, m) as a linear combination of Rα(u) of differntial orders; Rα(u) is Marcel Riesz's ultrahyperbolic kernel. We also obtain the following results: W?2k(u, m) = {□ + m2}kδ, k = 0, 1,…; W0(u, m) = δ; and {□ + m2}kW2k(u, m) = δ. Finally we prove that Wα(u, m = 0) = Rα(u). Several of these results, in the particular case µ = 1, were proved earlier by a completely different method. 相似文献
Given a graph G = (V, E), a set W í V{W \subseteq V} is said to be a resolving set if for each pair of distinct vertices u, v ? V{u, v \in V} there is a vertex x in W such that d(u, x) 1 d(v, x){d(u, x) \neq d(v, x)} . The resolving number of G is the minimum cardinality of all resolving sets. In this paper, conditions are imposed on resolving sets and certain conditional
resolving parameters are studied for honeycomb and hexagonal networks. 相似文献
The Dirichlet problem for elliptic systems of the second order with constant real and complex coefficients in the half-space k+ = {x = (x1,…,xk): xk > 0} is considered. It is assumed that the boundary values of a solution u = (u1,…,um) have the form ψ1ξ1 + · · · + ψnξn, 1 ≤ n ≤ m, where ξ1,· · ·,ξn is an orthogonal system of m-component normed vectors and ψ1,· · ·,ψn are continuous and bounded functions on ?k+. We study the mappings [C(?k+)]n ? (ψ1,…,ψn) → u(x) ? m and [C(?k+)]n ? (ψ1,…,ψn) → u(x) ? m generated by real and complex vector valued double layer potentials. We obtain representations for the sharp constants in inequalities between |u(x)| or |(z, u(x))| and ∥u|xk=0∥, where z is a fixed unit m-component vector, | · | is the length of a vector in a finite-dimensional unitary space or in Euclidean space, and (·,·) is the inner product in the same space. Explicit representations of these sharp constants for the Stokes and Lamé systems are given. We show, in particular, that if the velocity vector (the elastic displacement vector) is parallel to a constant vector at the boundary of a half-space and if the modulus of the boundary data does not exceed 1, then the velocity vector (the elastic displacement vector) is majorised by 1 at an arbitrary point of the half-space. An analogous classical maximum modulus principle is obtained for two components of the stress tensor of the planar deformed state as well as for the gradient of a biharmonic function in a half-plane. 相似文献