共查询到20条相似文献,搜索用时 46 毫秒
1.
T.A. Suslina 《Functional Analysis and Its Applications》2010,44(4):318-322
Homogenization in the small period limit for the solution ue of the Cauchy problem for a parabolic equation in Rd is studied. The coefficients are assumed to be periodic in Rd with respect to the lattice ɛG. As ɛ → 0, the solution u ɛ converges in L2(Rd) to the solution u0 of the effective problem with constant coefficients. The solution u ɛis approximated in the norm of the
Sobolev space H
1(Rd) with error O( ɛ); this approximation is uniform with respect to the L2-norm of the initial data and contains a corrector
term of order ɛ. The dependence of the constant in the error estimate on time t is given. Also, an approximation in H
1(Rd) for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained. 相似文献
2.
E. Di Benedetto 《Annali di Matematica Pura ed Applicata》1979,121(1):249-262
Sunto In questo lavoro si considera il problema di Cauchy per l'equazione di filtrazione ∂u/∂t=∂
2
ϕ(u)/∂x
2
nella regioneR×(0,T],0<T<∞. Sotto opportune ipotesi sulla funzione ϕ(u) si determina una stima dell'incremento temporale della soluzione u(x, t)
(intesa nel senso debole). Nel caso politropico (ϕ(u)=um), quando m>2 si trova in particolare un comportamento h?lderiano di u(x, t) rispetto a t con l'esponente1/(m−1); viene anche dimostrato che questo esponete è effettivamente assunto da una particolare soluzione, per cui la stima ottenuta
è la migliore possibile.
Entrata in Redazione il 18 luglio 1978. 相似文献
Entrata in Redazione il 18 luglio 1978. 相似文献
3.
We consider the problem −Δu=|u|
p−1u+λu in Ω with
on δΩ, where Ω is a bounded domain inR
N
,p=(N+2)/(N−2) is the critical Sobolev exponent,n the outward pointing normal and λ a constant. Our main result is that if Ω is a ball inR
N
, then for every λ∈R the problem admits infinitely many solutions. Next we prove that for every bounded domain Ω inR
3, symmetric with respect to a plane, there exists a constant μ>0 such that for every λ<μ this problem has at least one non-trivial
solution.
This work was supported by the Paris VI-Leiden exchange program
Supported by the Netherlands organisation for scientific research NWO, under number 611-306-016. 相似文献
4.
Walter Senn 《manuscripta mathematica》1991,71(1):45-65
We consider a variational problem with an integrandF:R
n
×R×R
n
→R that isZ-periodic in the firstn+1 variables and satisfies certain growth-conditions. By a recent result of Moser, there exist for every α∈R
n
minimal solutionsu:R
n
→R minimising ƒF(x, u(x), u
x
(x)) dx with respect to compactly supported variations ofu and such that sup |u(x)-αx|<∞. Given such a minimal solutionu we define the average action
(whereB
r
is ther-ball around 0∈R
n
) and show thatM(α) is indeed independent of the minimal solutionu satisfying sup |u(x)-αx|<∞. We prove that this average actionM(α) is strictly convex in α.
相似文献
5.
Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations 总被引:6,自引:0,他引:6
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(4):881-892
This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt + α1 uδ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x,t)∈ E R^2, and δu/δt + α2 δu/δx δ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x, t) ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0(x) ∈ H^s(R), and s ≥ 5/4 for the first equation and s≥301/108 for the second equation. 相似文献
6.
Tetsutaro Shibata 《Journal d'Analyse Mathématique》2007,102(1):347-358
We study the nonlinear Sturm-Liouville problem
where λ > 0 is an eigenvalue parameter and f(u) is a rapidly increasing function. For better understanding of the global behavior of the bifurcation branch in R+ × L
2(I), we establish precise asymptotic formulas up to the third term for the eigenvalue λ(α) associated with the eigenfunction
u
α with ‖u
α‖2 = α, as α → ∞. We show that there exists a new type of asymptotic formula for λ (α) as α → ∞. 相似文献
7.
N. G. Khoma 《Ukrainian Mathematical Journal》1998,50(12):1917-1923
In three spaces, we find exact classical solutions of the boundary-value periodic problem utt - a2uxx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator
and whose right-hand side is a nonlinear operator.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998. 相似文献
8.
An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed
convex proper cone inR
n and −Γ′ be the antipodes of the dual cone of Γ. Let
be a partial differential operator with constant coefficients inR
n, whereQ(ζ)≠0 onR
n−iΓ′ andP
i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R
n−iΓ′;P
j(ζ)=0, gradP
j(ζ)≠0} contains some real point on which gradP
j≠0 and gradP
j∉Γ∪(−Γ). LetC be an open cone inR
n−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in
{ξ∈R
n;P(ξ)=0}. Ifu∈ℒ′∩L
loc
2
(R
n−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition
implies that the support ofu is contained in Γ. 相似文献
9.
T. Shibata 《Annali di Matematica Pura ed Applicata》2007,186(3):525-537
We consider the nonlinear Sturm–Liouville problem
where λ > 0 is an eigenvalue parameter. To understand well the global behavior of the bifurcation branch in R
+ × L
2(I), we establish the precise asymptotic formula for λ(α), which is associated with eigenfunction u
α with ‖ u
α ‖2 = α, as α → ∞. It is shown that if for some constant p > 1 the function h(u) ≔ f(u)/u
p
satisfies adequate assumptions, including a slow growth at ∞, then λ(α) ∼ α
p−1
h(α) as α → ∞ and the second term of λ(α) as α → ∞ is determined by lim
u → ∞
uh′(u).
Mathematics Subject Classification (2000) 34B15 相似文献
(1) |
10.
The review article of Crandall, Ishii, and Lions [Bull. AMS,27, No. 1, 1–67 (1992)] devoted to viscosity solutions of first- and second-order partial differential equations contains the
exact Lax formula
for a solution to the Hamilton-Jacobi nonlinear partial differential equation
where the Cauchy datav:R
n
→R are chosen as a function properly convex and semicontinuous from below, ‖·‖=<·,·> is the usual norm inR
n
,n ∉Z
+, andt ∉R
+ is a positive evolution parameter. The article also states that there is no exact proof of the Lax formula (1) based on general
properties of the Hamiltonian-Jacobi equation (2). This work presents precisely such an exact proof of the Lax formula (1).
Pidstryhach Institute of Applied Problems in Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv; Courant Institute
of Mathematical Sciences at NYU, New York. Published in Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp.
20–25, 相似文献
((1)) |
((2)) |
11.
We investigate the linear periodic problem u
tt
−u
xx
=F(x, t), u(x+2π, t)=u(x, t+T)=u(x, t), ∈ ℝ2, and establish conditions for the existence of its classical solution in spaces that are subspaces of the Vejvoda-Shtedry
spaces.
Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308,
February, 1997. 相似文献
12.
We study a periodic problem for the equation u
tt−uxx=g(x, t), u(x, t+T)=u(x, t), u(x+ω, t)= =u(x, t), ℝ2 and establish conditions of the existence and uniqueness of the classical solution.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 558–565, April, 1997. 相似文献
13.
Let Ω, ⊂R
n and n ≥ 4 be even. We show that if a sequence {uj} in W1,n/2(Ω;R
n) is almost conformal in the sense that dist (∇uj,R
+SO(n)) converges strongly to 0 in Ln/2 and if uj converges weakly to u in W1,n/2, then u is conformal and ∇uj → ∇u strongly in L
loc
q
for all 1 < -q < n/2. It is known that this conclusion fails if n/2 is replaced by any smaller exponent p. We also prove
the existence of a quasiconvex function f(A) that satisfies 0 ≤ f(A) ≤ C (1 + |A|n/2) and vanishes exactly onR
+ SO(n). The proof of these results involves the Iwaniec-Martin characterization of conformal maps, the weak continuity and
biting convergence of Jacobians, and the weak-L1 estimates for Hodge decompositions. 相似文献
14.
Local and Global Existence of Solutions to
Initial Value Problems of Modified Nonlinear
Kawahara Equations 总被引:3,自引:0,他引:3
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(5):1035-1044
This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third +α the second partial dervative of u to x ,the second the third +β the third partial dervative of u to x ,the second the thire +γ the fifth partial dervative of u to x = 0,(x,t)∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo(x) ∈ H^s(R) with s ≥ 1/4, and a global solution exists if s ≥ 2. 相似文献
15.
Jan Persson 《Annali di Matematica Pura ed Applicata》1979,122(1):117-140
Summary Let Ω cR
n be an open set and let P be a linear partial differential operator with constant coefficients inR
n. Then Ω is said to be P-convex if for each f ε C∞(Ω) there is a u ε D′(Ω) such that P(D)u=f. A complete geometric characterization of P-convex sets inR
3 is given when P is of principal type and when Ω has C2-boundary. As a step in the proof one also obtains necessary and sufficient conditions for uniqueness in the local Cauchy
problem at simply characteristic points inR
3. The tools are a sophisticated use of the author's uniqueness cones on one hand and his semi-global nullsolutions on the
other hand. Hints are given on the difficulties that may be encountered inR
n for the same problem.
Entrata in Redazione il 7 giugno 1978. 相似文献
16.
Sergio Campanato 《Milan Journal of Mathematics》1990,60(1):113-131
Denoting byu a vector in R
N
defined on a bounded open set Ω ⊂ R
n
, we setH(u)={Dij u} and consider a basic differential operator of second ordera(H(u)) wherea(ξ) is a vector in R
N
, which is elliptic in the sense that it satisfies the condition (A). After a rapid comparison between this condition (A) and the classical definition of ellipticity, we shall prove that, if seu∈H
2 (Ω) is a solution of the elliptic systema(H(u))=0 in Ω thenH(u)∈H
loc
2, q
for someq>2. We then deduce from this the so called fundamental internal estimates for the matrixH(u) and for the vectorsDu andu.
We shall then present a first risult on h?lder regularity for the solutions of the system withf h?lder continuous in Ω, and a partial h?lder continuity risult for solutionsu∈H
2 (Ω) of a differential systema (x, u, Du, H (u))=b(x, u, Du) 相似文献
17.
Stephen D. Fisher 《Israel Journal of Mathematics》1977,28(1-2):129-140
Letg be a positive continuous function onR which tends to zero at −∞ and which is not integrable overR. The boundary-value problem −u″+g(u)=f, u′(±∞)=0, is considered forf∈L
1(R). We show that this problem can have a solution if and only ifg is integrable at −∞ and if this is so then the problem is solvable precisely when ∫
−∞
∞
. Some extensions of this result are also given.
Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation, Grant MPS
75-05501. 相似文献
18.
Givenμ, κ, c>0, we consider the functional
defined on allR
n
-valued functionsu on the open subset Ω ofR
n
which are smooth outside a free discontinuity setS
u, on which the tracesu
+,u
− on both sides have equal normal component (i.e.,u has a tangential jump alongS
u).E
Du=Eu − 1/3 (divu)I, withEu denoting the linearized strain tensor.
The functionalF is obtained from the usual strain energy of linearized elasticity by addition of a term (the second integral) which penalizes
the jump discontin uities of the displacement.
The lower semicontinuous envelope
is studied, with respect to theL
1 (Ω;R
n
)-topology, on the spaceP(Ω) of the functions of bounded deformation with distributional divergence inL
2(Ω) (F is extended with value +∞ on the wholeP(Ω)). The following integral representation is proved:
whereϕ is a convex function with linear growth at infinity. NowEu is a measure,ɛ
Du represents the density of the absolutely continuous part of the absolutely continuous part ofE
Du, whileE
s
D
u denotes the singular part and ϕ∞ the recession function ofϕ.
Finally, we show that
coincides with the functional which intervenes in the minimum problem for the displacement in the theory of Hencky’s plasticity
with Tresca’s yield conditions. 相似文献
19.
W. Ishizuka C. Y. Wang 《Calculus of Variations and Partial Differential Equations》2008,32(3):387-405
For a bounded domain Ω ⊂ R
n
endowed with L
∞-metric g, and a C
5-Riemannian manifold (N, h) ⊂ R
k
without boundary, let u ∈ W
1,2(Ω, N) be a weakly harmonic map, we prove that (1) u ∈ C
α (Ω, N) for n = 2, and (2) for n ≥ 3, if, in additions, g ∈ VMO(Ω) and u satisfies the quasi-monotonicity inequality (1.5), then there exists a closed set Σ ⊂ Ω, with H
n-2(Σ) = 0, such that for some α ∈ (0, 1).
C. Y. Wang Partially supported by NSF. 相似文献
20.
L. I. Danilov 《Theoretical and Mathematical Physics》1999,118(1):1-11
We prove the absolute continuity of the Dirac operator spectrum inR
2 with the scalar potential V and the vector potential A=(A1, A2) being periodic functions (with a common period lattice) such that V, Aj≠L
loc
q
(R
2), q>2.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 3–14, January, 1999. 相似文献