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1.
2.
Let S be the Schwartz space of rapidly decreasing real functions. The dual space S1 consists of the tempered distributions and the relation S ? L2(R) ? S1 holds. Let γ be the Gaussian white noise on S1 with the characteristic functional γ(ξ) = exp{?∥ξ∥2/2}, ξ ∈ S, where ∥·∥ is the L2(R)-norm. Let ν be the Poisson white noise on S1 with the characteristic functional ν(ξ) = exp?RR {[exp(iξ(t)u)] ? 1 ? (1 + u2)?1(iξ(t)u)} dη(u)dt), ξ ∈ S, where the Lévy measure is assumed to satisfy the condition ∫Ru2(u) < ∞. It is proved that γ1ν has the same dichotomy property for shifts as the Gaussian white noise, i.e., for any ω ∈ S1, the shift (γ1ν)ω of γ1ν by ω and γ1ν are either equivalent or orthogonal. They are equivalent if and only if when ω ∈ L2(R) and the Radon-Nikodym derivative is derived. It is also proved that for the Poisson white noice νω is orthogonal to ν for any non-zero ω in S1.  相似文献   

3.
We study the bifurcation problem ?Δu=g(u)+λ|?u|2+μ in Ω,u=0 on , where λ,μ?0 and Ω is a smooth bounded domain in RN. The singular character of the problem is given by the nonlinearity g which is assumed to be decreasing and unbounded around the origin. In this Note we prove that the above problem has a positive classical solution (which is unique) if and only if λ(a+μ)<λ1, where a=limt→+∞g(t) and λ1 is the first eigenvalue of the Laplace operator in H10(Ω). We also describe the decay rate of this solution, as well as a blow-up result around the bifurcation parameter. To cite this article: M. Ghergu, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  相似文献   

4.
5.
In this paper we prove a comparison result for weak solutions to linear elliptic problems of the type
?(aij(x)uxi)xj=f(x)?(x)inΩ,u=0on?Ω,
where Ω is an open set of Rn (n?2), ?(x)=(2π)?n/2exp(?|x|2/2), aij(x) are measurable functions such that aij(x)ξiξj??(x)|ξ|2 a.e. x∈Ω, ξ∈Rn and f(x) is a measurable function taken in order to guarantee the existence of a solution u∈H10(?,Ω) of (1.1). We use the notion of rearrangement related to Gauss measure to compare u(x) with the solution of a problem of the same type, whose data are defined in a half-space and depend only on one variable. To cite this article: M.F. Betta et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 451–456.  相似文献   

6.
Let α be a positive number. The one-dimensional viscoelastic problem
utt?uxx?αuxxt=f,x∈(?∞,0],t∈[0,+∞),
with unilateral boundary conditions
u(0,·)?0,(ux+αuxt)(0,·)?0,(u(ux+αuxt))(0,·)=0,
can be reduced to the following variational inequality:
λ11w=g+b,w?0,b?0,〈w,b〉=0.
Here λ?1(ω) is the causal determination of iω1+iαω. We show that the energy losses are purely viscous; this result is a consequence of the relation w?,b〉=0; since a priori, b is a measure and w? is defined only almost everywhere, this relation is not trivial. To cite this article: A. Petrov, M. Schatzman, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 983–988.  相似文献   

7.
Let Ω be a smooth bounded domain in RN. Assume fC1[0,∞) is a non-negative function such that f(u)/u is increasing on (0,∞). Let a be a real number and let b?0, b/≡0 be a continuous function such that b≡0 on . We study the logistic equation Δu+au=b(x)f(u) in Ω. The special feature of this work is the uniqueness of positive solutions blowing-up on , in a general setting that arises in probability theory. To cite this article: F.-C. C??rstea, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 447–452.  相似文献   

8.
We are concerned with the Lane–Emden–Fowler equation ?Δu=λf(u)+a(x)g(u) in Ω, subject to the Dirichlet boundary condition u=0 on ?Ω, where Ω?RN is a smooth bounded domain, λ is a positive parameter, a:Ω→[0,∞) is a Hölder function, and f is a positive nondecreasing continuous function such that f(s)/s is nonincreasing in (0,∞). The singular character of the problem is given by the nonlinearity g which is assumed to be unbounded around the origin. In this Note we discuss the existence and the uniqueness of a positive solution of this problem and we also describe the precise decay rate of this solution near the boundary. The proofs rely essentially on the maximum principle and on elliptic estimates. To cite this article: M. Ghergu, V.D. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).  相似文献   

9.
We study degeneration for ? → + 0 of the two-point boundary value problems
τ?±u := ?((au′)′ + bu′ + cu) ± xu′ ? κu = h, u(±1) = A ± B
, and convergence of the operators T?+ and T?? on L2(?1, 1) connected with them, T?±u := τ?±u for all
u?D(T?±, D(T?±) := {u ? L2(?1, 1) ∣ u″ ? L2(?1, 1) &; u(?1) = u(1) = O}, T0+u: = xu′
for all
u?D(TO+), D(TO+) := {u ? L2(?1, 1) ∣ xu′ ? L2(?1, 1) &; u(?1) = u(1) = O}
. Here ? is a small positive parameter, λ a complex “spectral” parameter; a, b and c are real b-functions, a(x) ? γ > 0 for all x? [?1, 1] and h is a sufficiently smooth complex function. We prove that the limits of the eigenvalues of T?+ and of T?? are the negative and nonpositive integers respectively by comparison of the general case to the special case in which a  1 and bc  0 and in which we can compute the limits exactly. We show that (T?+ ? λ)?1 converges for ? → +0 strongly to (T0+ ? λ)?1 if R e λ > ? 12. In an analogous way, we define the operator T?+, n (n ? N in the Sobolev space H0?n(? 1, 1) as a restriction of τ?+ and prove strong convergence of (T+?,n ? λ)?1 for ? → +0 in this space of distributions if R e λ > ?n ? 12. With aid of the maximum principle we infer from this that, if h?C1, the solution of τ?+u ? λu = h, u(±1) = A ± B converges for ? → +0 uniformly on [?1, ? ?] ∪ [?, 1] to the solution of xu′ ? λu = h, u(±1) = A ± B for each p > 0 and for each λ ? C if ? ?N.Finally we prove by duality that the solution of τ??u ? λu = h converges to a definite solution of the reduced equation uniformly on each compact subset of (?1, 0) ∪ (0, 1) if h is sufficiently smooth and if 1 ? ?N.  相似文献   

10.
We show that for every u∈BV(Ω;S1), there exists a bounded variation function ?∈BV(Ω;R) such that u=ei? a.e. on Ω and |?|BV?2|u|BV. The constant 2 is optimal in dimension n>1. To cite this article: J. Dávila, R. Ignat, C. R. Acad. Sci. Paris, Ser. I 337 (2003).  相似文献   

11.
12.
R. Pellikaan (Arithmetic, Geometry and Coding Theory, Vol.  4, Walter de Gruyter, Berlin, 1996, pp. 175–184) introduced a two variable zeta-function Z(t,u) for a curve over a finite field Fq which, for u=q, specializes to the usual zeta-function and he proved rationality: Z(t,u)=(1?t)?1(1?ut)?1P(t,u) with P(t,u)∈Z[t,u]. We prove that P(t,u) is absolutely irreducible. This is motivated by a question of J. Lagarias and E. Rains about an analogous two variable zeta-function for number fields. To cite this article: N. Naumann, C. R. Acad. Sci. Paris, Ser. I 336 (2003).  相似文献   

13.
We characterize in terms of monotonicity basic properties of quasilinear elliptic partial differential operators which make it possible to obtain a Liouville-type comparison principle for entire solutions of quasilinear elliptic partial differential inequalities of the form A(u)+|u|q?1u?A(v)+|v|q?1v, which belong only locally to the corresponding Sobolev spaces on Rn,n?2. We establish that such properties are inherent for a wide class of quasilinear elliptic partial differential operators. Typical examples of such operators are the p-Laplacian and its well-known modifications for 1<p?2. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 336 (2003).  相似文献   

14.
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 ¦▽m2 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   

15.
We investigate the boundary value problem ?u?t = ?2u?x2 + u(1 ? u ? rv), ?v?t = ?2v?x2 ? buv, u(?∞, t) = v(∞, t) = 0, u(∞, t) = 1, and v(?∞, t) = γ ?t > 0 where r > 0, b > 0, γ > 0 and x?R. This system has been proposed by Murray as a model for the propagation of wave fronts of chemical activity in the Belousov-Zhabotinskii chemical reaction. Here u and v are proportional to the concentrations of bromous acid and bromide ion, respectively. We determine the global stability of the constant solution (u, v) ≡ (1,0). Furthermore we introduce a moving coordinate and for each fixed x?R we investigate the asymptotic behavior of u(x + ct, t) and v(x + ct, t) as t → ∞ for both large and small values of the wave speed c ? 0.  相似文献   

16.
We consider the problem of minimizing integral functionals of the form I(u) = ∝Ω F(x, ▽[k]u(x)) dx, where Ω ?Rp, u:ω →R 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   

17.
This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gi(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when φ(x) = max{f(x, y) ¦ y ? Ωy}, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form miny ? Ωxmaxy ? Ωyf(x, y) under the assumption that Ωmξ={x|gi(x)?0, j=1,…,s} ∩Rn, Ωy={y|ζi(y)?0, i-1,…,t} ∩ Rm, with f, gj, ζi continuously differentiable, f(x, ·) concave, ζi convex for i = 1,…, t, and Ωx, Ωy compact.  相似文献   

18.
In this paper we prove a global well-posedness result for the following Cauchy problem:
?ttu?Δu+a0?tu+i=13ai?xiu+Vu=?u|u|α?1,for(t,x)∈Rt×R3x,u(0)=f,?tu(0)=g,
where the initial data (f,g)∈H?1(R3)×L2(R3) are compactly supported, 1?α<5, ai(t,x)∈L(Rt×Rx3), V(t,x)∈L(Rt;L3(R3x)). To cite this article: N. Visciglia, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  相似文献   

19.
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 (u, α · pu) = (u, r(?q?r)u) is shown to hold, u being an eigenfunction of the operator
Hu = TU : = (α · p + β + q)u
,
D(H) = {u ¦ u ∈ [Hloc1(R+3)]4, r?12u, TU ∈ [L2(R)3]4}
(R+3 := R?{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   

20.
We consider the Helmholtz equation with a variable index of refraction n(x), which is not necessarily constant at infinity but can have an angular dependency like n(x)→n(x/|x|) as |x|→∞. We prove that the Sommerfeld condition at infinity still holds true under the weaker form
1R|x|?R?u?in1/2x|x|ux|x|2dx→0,asR→∞.
Our approach consists in proving this estimate in the framework of the limiting absorbtion principle. We use Morrey–Campanato type of estimates and a new inequality on the energy decay, namely
Rd?n(ω)2|u|2|x|dx?C,ω=x|x|.
It is a striking feature that the index n appears in this formula and not the phase gradient, in apparent contradiction with existing literature. To cite this article: B. Perthame, L. Vega, C. R. Acad. Sci. Paris, Ser. I 337 (2003).  相似文献   

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