共查询到20条相似文献,搜索用时 31 毫秒
1.
Thomas I Seidman 《Journal of Differential Equations》1985,60(2):151-173
We obtain a strict coercivity estimate, (generalizing that of T. I. Seidman [J. Differential Equations 19 (1975), 242–257] in considering spatial variation) for second order elliptic operators A: u ? ?▽ · γ(·, ▽u) with γ “radial in the gradient” ?γ(·, ξ) = a(·, |ξ|)ξ for ξ ? m. The estimate is then applied to obtain existence of solutions of boundary value problems: with Dirichlet conditions. 相似文献
2.
Wolf von Wahl 《Journal of Functional Analysis》1978,27(1):118-135
This paper deals with classical solvability for all t of semilinear parabolic equations u′ + A(t)u = f(t, x, u, ▽u, …, ▽2m ? 1u). It is shown that the right side is allowed to grow faster than in ▽mu if a Hölder norm of u is known a priori. In the second part an example is given where an a priori estimate of a Hölder norm of u is available. Moreover, we give a new maximum principle. 相似文献
3.
Tomas Schonbek 《Journal of Differential Equations》1985,56(2):290-296
New and more elementary proofs are given of two results due to W. Littman: (1) Let . The estimate cannot hold for all u?C0∞(Q), Q a cube in , some constant C. (2) Let n ? 2, p ≠ 2. The estimate cannot hold for all C∞ solutions of the wave equation □u = 0 in ; all t ?; some function C: → . 相似文献
4.
The asymptotic behaviour as t tends to +∞ of the solution of in N × +, p > 1, was studied. It was proved that the behaviour depends strongly on the sign of and also on the rate of decay of the admissible initial data u(0, x) as tends to +∞. 相似文献
5.
A.G Ramm 《Journal of Mathematical Analysis and Applications》1984,98(1):92-98
Consider the exterior boundary value problem (▽2 + K2) u = 0, in Ω, k >0. , where Γ is a smooth closed connected surface in 3, , ∝ is called the radiation pattern. We prove that when h runs through any dense set in L2(Γ) the corresponding radiation pattern ∝(k,n) runs through a dense set in L2(S2) for any k >0, where S2 is the unit sphere in 3. 相似文献
6.
Hans Engler 《Journal of Functional Analysis》1985,64(3):412-435
Consider the heat equation ?ru ? Δxu = 0 in a cylinder Ω × [0,T] ? n+1 smooth lateral boundary under zero Neumann or Dirichlet conditions. Geometric conditions for Ω are given that guarantee that for a given P, 6▽xu(·, t)6Lp will be non-increasing for any solution. Decay rates are also given. For arbitrary Ω and p, it is shown how to construct an equivalent Lp-norm, such that ▽x(·, t) is non-increasing in this norm. 相似文献
7.
We consider the problem of minimizing integral functionals of the form , where Ω ?p, u:ω → and ▽[k]u denotes the set of all partial derivatives of u with orders ?k. The method is based on a characterization of null Lagrangians L(▽ku) depending only on derivatives of order k. Applications to elasticity and other theories of mechanics are given. 相似文献
8.
Hartmut Pecher 《Journal of Functional Analysis》1985,63(1):101-122
The scattering operator which belongs to a pair of PDEs consisting of the Klein-Gordon equation and a perturbation of it by a power-like nonlinearity z.hfl;(u) is studied. It is shown that this operator can be defined on a whole neighbourhood of the origin in energy space if , where and the space dimension n ? 2 is arbitrary. 相似文献
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Yueh-er Kuo 《Journal of Mathematical Analysis and Applications》1976,56(2):346-350
Let be column-wise partitioned matrices over complex numbers. Then an extended Kronecker product is , where Ai ? Bi is the Kronecker product of Ai and Bi. Some properties of an extended Kronecker product of matrices are investigated. The properties of the solutions of the systems of linear equations whose coefficient matrices are extended Kronecker products of matrices are studied. 相似文献
12.
Hubert Kalf 《Journal of Functional Analysis》1976,21(4):389-396
For a class of potentials including the Coulomb potential q = μr?1 with ¦ μ ¦ < 1 (1) (i.e., atomic numbers Z ? 137), the virial theorem is shown to hold, u being an eigenfunction of the operator , (+3 := ?{0}). The result implies in particular that H with (1) does not have any eigenvalues embedded in the continuum. The proof uses a scale transformation. 相似文献
13.
Gustavo Perla Menzala 《Journal of Mathematical Analysis and Applications》1983,93(2):385-396
The semilinear wave equation in , is studied where □ denotes the d'Alembertian operator and 1 means spatial convolution. Under mild assumptions on the real-valued function V and 2 ? p ? 3 the well-posedness of the Cauchy problem is proved. Furthermore, some properties of the solutions of the equation are analyzed such as the asymptotic behavior of local energy as in the case of zero mass. Our results extend that of Perla Menzala and Strauss, where case p = 2 was studied. 相似文献
14.
In a recent paper [3] the authors derived maximum principles which involved , where u(x) is a classical solution of an alliptic differential equation of the form (. In this paper these results are extended to the more general case in which is replaced by h(u, q2). 相似文献
15.
An elastic-plastic bar with simply connected cross section Q is clamped at the bottom and given a twist at the top. The stress function u, at a prescribed cross section, is then the solution of the variational inequality (0.1) is equal to the angle of the twist (after normalizing the units). Introducing the Lagrange multiplier λθ1, the unloading problem consists in solving the variational inequality (0.3) is the twisting angle for the unloaded bar; θ2 < θ1. Let (0.4) , and denote by the solutions of (0.1), (0.3), respectively, when K is replaced by . The following results are well known for the loading problem (0.1):(0.5) ; (0.6) the plastic set is connected to the boundary. In this paper we show that, in general, (0.7) ; (0.8) the plastic set is not connected to the boundary. That is, we construct domains Q for which (0.7) and (0.8) hold for a suitable choice of θ1, θ2. 相似文献
16.
Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem . Here the Aj's are constant, k × k Hermitian matrices, x = (x1,…, xn), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit , where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U0(t) and such that depending on ? and that the local energy of nonstatic solutions decays as . More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation at infinity. 相似文献
17.
Walter Rudin 《Journal of Functional Analysis》1983,50(1):100-126
Let B be the open unit ball of n, n > 1. Let I (for “inner”) be the set of all u ? H °(B) that have a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ? I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ? X, y ? Y, and let be the norm closure, in L∞ = L∞(S), of X. Some results: set I is dense in the unit ball of H∞(B) in the compact-open topology. On is weak1-dense in does not contain . (When .) Every unimodular is a pointwise limit a.e. of products . The zeros of every in the ball algebra (but not of every H∞-function) can be matched by those of some u ? I, as can any finite number of derivatives at 0 if . However, cannot be bounded in B if u ? I is non-constant. 相似文献
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19.
Philip Brenner 《Journal of Differential Equations》1985,56(3):310-344
Asymptotic properties of solutions of the nonlinear Klein-Gordon equation ?t2u ? Δu + m2u + f(u) = 0 (NLKG) , , are investigated, which are inherited from the corresponding solutions v of the (linear) Klein-Gordon equation ?t2v ? Δv + m2v = 0, , (KG) In particular, the finiteness of time-integrals in Lq over R+ of certain Sobolevnorms in space of the solution is proved to be such a hereditary property. Together with a device by W. A. Strauss and a weak decay result for the (KG) due to R. S. Strichartz, this is used to prove that under suitable restrictions on the nonlinearity, the scattering operator for the (NLKG) is defined on all of L21 × L2 for n = 3. 相似文献