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1.
The Blow-up Locus of Heat Flows for Harmonic Maps 总被引:5,自引:0,他引:5
Abstract
Let M and N be two compact Riemannian manifolds. Let u
k
(x, t) be a sequence of strong stationary weak heat flows from M×R
+ to N with bounded energies. Assume that u
k→u weakly in H
1, 2(M×R
+, N) and that Σt is the blow-up set for a fixed t > 0. In this paper we first prove Σt is an H
m−2-rectifiable set for almost all t∈R
+. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the
limiting map u is also a strong stationary weak heat flow, Σt is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly
quasi-harmonic sphere and a blow-up set ∪t<0Σt× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.
This work is supported by NSF grant 相似文献
2.
James Olsen 《Israel Journal of Mathematics》1972,11(1):1-13
The principal result of this paper is that the convex combination of two positive, invertible, commuting isometries ofL
p(X,F, μ) 1<p<+∞, one of which is periodic, admits a dominated estimate with constantp/p−1. In establishing this, the following analogue of Linderholm’s theorem is obtained: Let σ and ε be two commuting non-singular
point transformations of a Lebesgue Space with τ periodic. Then given ε>O, there exists a periodic non-singular point transformation σ′ such that σ′ commutes with τ and μ(x:σ′x≠σx}<ε. Byan approximation argument, the principal result is applied to the convex combination of two isometries ofL
p (0, 1) induced by point transformations of the form τx=x
k,k>0 to show that such convex combinations admit a dominated estimate with constantp/p−1.
Research supported in part by NSF Grant No. GP-7475. A portion of the contents of this paper is based on the author’s doctoral
dissertation written under the direction of Professor R. V. Chacon of the University of Minnesota. 相似文献
3.
In this article we consider a pq-dimensional random vector x distributed normally with mean vector θ and covariance matrix Λ assumed to be positive definite. On the basis of N independent observations on the random vector x, we want to estimate parameters and test the hypothesis H: Λ = Ψ ⊗ Σ, where Ψ = (ψ
ij
): q × q, ψ
qq
= 1, and Σ = (σ
ij
): p × p, and Λ = (ψ
ij
Σ), the Kronecker product of Ψ and Σ. That is instead of 1/2pq(pq + 1) parameters, it has only 1/2p(p + 1) + 1/2q(q + 1) − 1 parameters. A test based on the likelihood ratio is given to check if this model holds. And, when this model holds,
we test the hypothesis that Ψ is a matrix with intraclass correlation structure. The maximum likelihood estimators (MLE) are
obtained under the hypothesis as well as under the alternatives. Using these estimators the likelihood ratio tests (LRT) are
obtained. One of the main objects of the paper is to show that the likelihood equations provide unique estimators.
相似文献
4.
Shu-Yu Hsu 《Mathematische Annalen》2003,325(4):665-693
We prove that the solution u of the equation u
t
=Δlog u, u>0, in (Ω\{x
0})×(0,T), Ω⊂ℝ2, has removable singularities at {x
0}×(0,T) if and only if for any 0<α<1, 0<a<b<T, there exist constants ρ0, C
1, C
2>0, such that C
1
|x−x
0|α≤u(x,t)≤C
2|x−x
0|−α holds for all 0<|x−x
0|≤ρ0 and a≤t≤b. As a consequence we obtain a sufficient condition for removable singularities at {∞}×(0,T) for solutions of the above equation in ℝ2×(0,T) and we prove the existence of infinitely many finite mass solutions for the equation in ℝ2×(0,T) when 0≤u
0∉L
1
(ℝ2) is radially symmetric and u
0L
loc
1(ℝ2).
Received: 16 December 2001 / Revised version: 20 May 2002 / Published online: 10 February 2003
Mathematics Subject Classification (1991): 35B40, 35B25, 35K55, 35K65 相似文献
5.
We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain Ω of R3. We first prove the local existence of solutions (ρ,u) in C([0,T*]; (ρ∞ +H3(Ω)) × under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the
velocity u in t>0, we conclude that (ρ,u) is a classical solution in (0,T**)×Ω for some T** ∈ (0,T*]. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of Ω. 相似文献
6.
In this paper, we study the initial-boundary value problem of the porous medium equation u
t
= Δu
m
+ V(x)u
p
in a cone D = (0, ∞) × Ω, where V(x) ~ (1 + |x|)
σ
. Let ω
1 denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on Ω and let l denote the positive root of l
2 + (n − 2)l = ω
1. We prove that if m ≤ p ≤ m + (2 + σ)/(n + l), then the problem has no global nonnegative solutions for any nonnegative u
0 unless u
0 = 0; if p > m + (2 + σ)/n, then the problem has global solutions for some u
0 ≥ 0. 相似文献
7.
Yehoram Gordon 《Israel Journal of Mathematics》1969,7(2):151-163
Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ
p(X) as inf{Σ
i
=1/m
|x*(x
i)|p
p Σ
i
=1/m
‖x
i‖p
p]1
p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x
1,x
2, …,x
m} ⊂X such that Σ
i
=1/m
‖x
i‖>0. It follows immediately from [2] thatμ
p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ
p(X) for various spaces, and obtain some asymptotic estimates ofμ
p(X) for general finite dimensional Banach spaces.
This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof.
A. Dvoretzky and Prof. J. Lindenstrauss. 相似文献
8.
We study the existence and the properties of reduced measures for the parabolic equations ∂
t
u − Δu + g(u) = 0 in Ω × (0, ∞) subject to the conditions (P): u = 0 on ∂Ω × (0, ∞), u(x, 0) = μ and (P′): u = μ′ on ∂Ω × (0, ∞), u(x, 0) = 0, where μ and μ′ are positive Radon measures and g is a continuous nondecreasing function. 相似文献
9.
We present the bi-Hamiltonian structure and Lax pair of the equation ρt = bux+(1/2)[(u
2
−ux
2
)ρ]x, where ρ = u − uxx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show
that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions. 相似文献
10.
Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤd, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant,
and ξ = {ξ(x): x∈ℤ
d
} is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate,
then the solution u is asymptotically intermittent.
In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the
vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e
s
/θ] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result
is that, for fixed x, y∈ℤ
d
and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w
ρ∥−2
ℓ2Σz ∈ℤd
w
ρ(x+z)w
ρ(y+z). In this expression, ρ = θ/κ while w
ρ:ℤd→ℝ+ is given by w
ρ = (v
ρ)⊗
d
with v
ρ: ℤ→ℝ+ the unique centered ground state (i.e., the solution in ℓ2(ℤ) with minimal l
2-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞).
empty
It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation
coefficient of u(x, t) and u(y, t) converges to δ
x, y
(resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation
structure.
Received: 5 March 1997 / Revised version: 21 September 1998 相似文献
11.
Summary For P∈ F2[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σn≧0 p(A,n)zn ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ d|n, d∈A d, namely, the periodicity of the sequences (σ(A,2kn) mod 2k+1)n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula
to σ(A,2kn) mod 2k+1 will be established, from which it will be shown that the weight σ(A1,2kzi) mod 2k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$
is moved on some other orbit zj when A1 is replaced by A2 with A1= A(P1) and A2= A(P2) P1 and P2 being irreducible in F2[z] of the same odd order. 相似文献
12.
We consider the existence and uniqueness of singular solutions for equations of the formu
1=div(|Du|p−2
Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2.
Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the
existence of very singular solutions, i.e. singular solutions such that ∫|x|≤r
u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result.
In the case ϕ(u)=u
q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal.
Dedicated to Professor Shmuel Agmon 相似文献
13.
On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary 总被引:1,自引:0,他引:1
We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu
t=uxx, ut=(um)xxand
(m>1) forx>0,t>0 with nonlinear boundary conditions−u
x=up,−(u
m)x=upand
forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove
that for each problem there exist positive critical valuesp
0,pc(withp
0<pc)such that forp∃(0,p
0],all solutions are global while forp∃(p0,pc] any solutionu≢0 blows up in a finite time and forp>p
csmall data solutions exist globally in time while large data solutions are nonglobal. We havep
c=2,p
c=m+1 andp
c=2m for each problem, whilep
0=1,p
0=1/2(m+1) andp
0=2m/(m+1) respectively.
This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications
at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210. 相似文献
14.
Let π = (d
1, d
2, ..., d
n
) and π′ = (d′
1, d′
2, ..., d′
n
) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π ⊲ π′, if and only if π ≠ π′, Σ
i=1
n
d
i
= Σ
i=1
n
d′
i
, and Σ
i=1
j
d
i
≤ Σ
i=1
j
d′
i
for all j = 1, 2, ..., n. Weuse C
π
to denote the class of connected graphs with degree sequence π. Let ρ(G) be the spectral radius, i.e., the largest eigenvalue of the adjacent matrix of G. In this paper, we extend the main results of [Liu, M. H., Liu, B. L., You, Z. F.: The majorization theorem of connected
graphs. Linear Algebra Appl., 431(1), 553–557 (2009)] and [Bıyıkoğlu, T., Leydold, J.: Graphs with given degree sequence and maximal spectral radius. Electron. J. Combin., 15(1), R119 (2008)]. Moreover, we prove that if π and π′ are two different non-increasing degree sequences of unicyclic graphs with π ⊲ π′, G and G′ are the unicyclic graphs with the greatest spectral radii in C
π
and C′
π
, respectively, then ρ(G) < ρ(G′). 相似文献
15.
Jorge García-Melián José C. Sabina De Lis Julio D. Rossi 《NoDEA : Nonlinear Differential Equations and Applications》2007,14(5-6):499-525
We deal with positive solutions of Δu = a(x)u
p
in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ1 where, roughly speaking, σ1 is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile
of the solution as λ → σ1.
Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D.
Rossi). J.D. Rossi is a member of CONICET. 相似文献
16.
Christine Lescop 《Inventiones Mathematicae》1998,133(3):613-681
We study the following question: How does the Casson-Walker invariant λ of a rational homology 3-sphere obtained by gluing
two pieces along a surface depend on the two pieces? Our partial answer may be stated as follows. For a compact oriented 3-manifold
A with boundary ∂A, the kernel L
A
of the map from H
1(∂A;Q) to H
1(A;Q) induced by the inclusion is called the Lagrangian of A. Let Σ be a closed oriented surface, and let A, A′, B and B′ be four rational homology handlebodies such that ∂A, ∂A′, −∂B and −∂B′ are identified via orientation-preserving homeomorphisms with Σ. Assume that L
A
= L
A
′ and L
B
= L
B
′ inside H
1(Σ;Q) and also assume that L
A
and L
B
are transverse. Then we express
in terms of the form induced on ∧3 L
A
by the algebraic intersection on H
2(A∪Σ−A′) paired to the analogous form on ∧3 L
B
via the intersection form of Σ. The simple formula that we obtain naturally extends to the extension of the Casson-Walker
invariant of the author. It also extends to gluings along non-connected surfaces.
Oblatum 6-III-1995 & 31-X-1997 相似文献
17.
Simon [J. Approxim. Theory,
127, 39–60 (2004)] proved that the maximal operator σα,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H
p
to the space L
p
for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σα,κ,* is bounded from the martingale Hardy space H
1/(1+α) to the space weak- L
1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H
p
such that the maximal operator σα,κ,*
f does not belong to the space L
p
. 相似文献
18.
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) |
19.
Abraham Neyman 《Israel Journal of Mathematics》1984,48(2-3):129-138
For fixed 1≦p<∞ theL
p-semi-norms onR
n
are identified with positive linear functionals on the closed linear subspace ofC(R
n
) spanned by the functions |<ξ, ·>|
p
, ξ∈R
n
. For every positive linear functional σ, on that space, the function Φσ:R
n
→R given by Φσ is anL
p-semi-norm and the mapping σ→Φσ is 1-1 and onto. The closed linear span of |<ξ, ·>|
p
, ξ∈R
n
is the space of all even continuous functions that are homogeneous of degreep, ifp is not an even integer and is the space of all homogeneous polynomials of degreep whenp is an even integer. This representation is used to prove that there is no finite list of norm inequalities that characterizes
linear isometric embeddability, in anyL
p unlessp=2.
Supported by the National Science Foundation MCS-79-06634 at U.C. Berkeley. 相似文献
20.
Motivated by problems arising from Arithmetic Geometry, in an earlier article one of the authors studied germs of holomorphic
isometries between bounded domains with respect to the Bergman metric. In the case of a germ of holomorphic isometry f: (Δ, λ ds
Δ2;0) → (Ω, ds
Ω2;0) of the Poincaré disk Δ into a bounded symmetric domain Ω ⋐ ℂ
N
in its Harish-Chandra realization and equipped with the Bergman metric, f extends to a proper holomorphic isometric embedding F: (Δ, λ ds
Δ2;) → (Ω, ds
Ω2) and Graph(f) extends to an affine-algebraic variety V ⊂ ℂ × ℂ
N
. Examples of F which are not totally geodesic have been constructed. They arise primarily from the p-th root map ρ
p
: H → H
p
and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H
3 of genus 3. In the current article on the one hand we examine second fundamental forms σ of these known examples, by computing explicitly φ = |σ|2. On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincaré disk into polydisks. For such mappings expressing via the inverse Cayley
transform in terms of the Euclidean coordinate τ = s + it on the upper half-plane H, we have φ(τ) = t
2
u(τ), where u|
t=0 ≢ 0. We show that u must satisfy the first order differential equation δu/δt|
t=0 ≡ 0 on the real axis outside a finite number of points at which u is singular. As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the
Poincaré disk into the polydisk must develop singularities along the boundary circle. The equation δu/δt|
t=0 ≡ 0 along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries
into Siegel upper half-planes arising from G. Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical
and the computational results of the article. 相似文献