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1.
We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable t and periodic conditions with respect to coordinates x
1,..., x
p for a nonisotropic (concerning differentiation with respect to t and x
1,..., x
p) partial differential equation with constant complex coefficients. We establish conditions for the existence and uniqueness of a solution of this problem and prove metric theorems on lower bounds for small denominators appearing in the course of the construction of its solution. 相似文献
2.
We consider the existence and uniqueness of singular solutions for equations of the form u
1=div(| Du| p−2
Du)-φ u), with initial data u( x, 0)=0 for x⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 and p>2.
Under a growth condition on ϕ( u) as u→∞, (H1), we prove that for every c>0 there exists a singular solution such that u( x, t)→ cδ( x) as t→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→∞ as t→0. Finally, for functions ϕ which behave like a power for large u 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 for q< p*= p-1+( p/N) and very singular solutions for p-1< q< p*. These ranges are optimal.
Dedicated to Professor Shmuel Agmon 相似文献
3.
Let f be a non-decreasing C 1-function such that
and F( t)/ f
2
a( t)→ 0 as t → ∞, where F( t)=∫
0
t
f( s) d s and a ∈ (0, 2]. We prove the existence of positive large solutions to the equation Δu + q( x)|Δ u|
a
= p( x) f( u) in a smooth bounded domain Ω ⊂R N, provided that p, q are non-negative continuous functions so that any zero of p is surrounded by a surface strictly included in Ω on which p is positive. Under additional hypotheses on p we deduce the existence of solutions if Ω is unbounded. 相似文献
4.
We establish conditions for the existence and uniqueness of a solution of a problem with multipoint conditions with respect to a selected variable t (in the case of multiple nodes) and periodic conditions with respect to x
1,..., x
p for a nonisotropic partial differential equation with constant complex coefficients. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of this problem. 相似文献
5.
We consider a variant of the classical problem of finding the size of the largest cap in the r-dimensional projective geometry PG( r, 3) over the field IF 3 with 3 elements. We study the maximum size f(n) of a subset S of IF
3
n
with the property that the only solution to the equation x
1+x 2+x 3= 0 is x
1=x 2=x 3. Let c
n=f(n) 1/n and c=sup{ c
1, c 2, ...}. We prove that c>2.21, improving the previous lower bound of 2.1955 ... 相似文献
7.
We study the behaviour of the positive solutions to the Dirichlet problem IR
n
in the unit ball in IR
R
where p<( N+2)/( N−2) if N≥3 and λ varies over IR. For a special class of functions g viz., g( x)= u
0
p
( x) where u
0 is the unique positive solution at λ=0, we prove that for certain λ’s nonradial solutions bifurcate from radially symmetric
positive solutions. When N=1, we obtain the complete bifurcation diagram for the positive solution curve. 相似文献
8.
We consider nonnegative solutions of initial-boundary value problems for parabolic equations u
t=u xx, u t=(u m) xxand
( m>1) for x>0, t>0 with nonlinear boundary conditions− u
x=u p,−( u
m) x=u pand
for x=0, t>0, where p>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 values p
0,p c(with p
0<p c)such that for p∃(0, p
0],all solutions are global while for p∃(p 0,p c] any solution u≢0 blows up in a finite time and for p>p
csmall data solutions exist globally in time while large data solutions are nonglobal. We have p
c=2, p
c=m+1 and p
c=2m for each problem, while p
0=1, p
0=1/2(m+1) and p
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
9.
First we prove the existence of a nontrivial smooth solution for a p-Laplacian equation with a ( p − 1)-linear nonlinearity and a noncoercive Euler functional, under hypotheses including resonant problems with respect to
the principal eigenvalue of (-D p, W1,p0( Z)){(-{\it \Delta}_p,\,W^{1,p}_0(Z))} . Then, for the semilinear problem (i.e., p = 2), assuming nonuniform nonresonance at infinity and zero, we prove a multiplicity theorem which provides the existence
of at least three nontrivial solutions, two being of opposite constant sign. Our approach combines minimax techniques with
Morse theory and truncation arguments. 相似文献
10.
We consider the partition function Z(N; x
1
, …, x N, y
1
, …, y N) of the square ice model with domain wall boundary conditions. We give a simple proof that Z is symmetric with respect to
all its variables when the global parameter a of the model is set to the special value a = e iπ/3
. Our proof does not use any determinant interpretation of Z and can be adapted to other situations (e.g., to some symmetric
ice models). 相似文献
11.
Summary We consider the d-dimensional Bernoulli bond percolation model and prove the following results for all p
c
: (1) The leading power-law correction to exponential decay of the connectivity function between the origin and the point (L, 0, ..., 0) isL
–(d–1)/2
. (2) The correlation length, (p) is real analytic. (3) Conditioned on the existence of a path between the origin and the point (L, 0, ..., 0), the hitting distribution of the cluster in the intermediate planes,x
1
=qL,0, obeys a multidimensional local limit theorem. Furthermore, for the two-dimensional percolation system, we prove the absence of a roughening transition: For allp>p
c
, the finite-volume conditional measures, defined by requiring the existence of a dual path between opposing faces of the boundary, converge—in the infinite-volume limit—to the standard Bernoulli measure.Work supported in part by G.N.A.F.A. (C.N.R.)Work supported in part by NSF Grant No. DMS-88-06552 相似文献
12.
We study the degenerate parabolic equation t∂u= a( δ( x)) upΔ u− g( u) in Ω×(0,∞), where Ω⊂ RN ( N?1) is a smooth bounded domain, p?1, δ( x)=dist( x,∂ Ω) and a is a continuous nondecreasing function such that a(0)=0. Under some suitable assumptions on a and g we prove the existence and the uniqueness of a classical solution and we study its asymptotic behavior as t→∞. 相似文献
13.
Let a1, a2, . . . , am ∈ ℝ 2, 2≤ f ∈ C([0,∞)), gi ∈ C([0,∞)) be such that 0≤ gi( t)≤2 on [0,∞) ∀ i=1, . . . , m. For any p>1, we prove the existence and uniqueness of solutions of the equation ut=Δ(log u), u>0, in satisfying and log u( x, t)/log| x|→− f( t) as | x|→∞, log u( x, t)/log| x− ai|→− gi( t) as | x− ai|→0, uniformly on every compact subset of (0, T) for any i=1, . . . , m under a mild assumption on u0 where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ 2 with prescribed singularities at a finite number of points in the domain. 相似文献
14.
Let K be a field of characteristic zero, n ≥ 5 an integer, f( x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, the ring of integers in the pth cyclotomic field, C
f, p
: y
p
= f( x) the corresponding superelliptic curve and J( C
f, p
) its jacobian. Assuming that either n = p + 1 or p does not divide n( n − 1), we prove that the ring of all endomorphisms of J( C
f, p
) coincides with . The same is true if n = 4, the Galois group of f( x) is the full symmetric group S
4 and K contains a primitive pth root of unity.
An erratum to this article can be found at 相似文献
15.
In this article, we discuss the blow-up problem of entire solutions of a class of second-order quasilinear elliptic equation Δ p u ≡ div(|? u| p?2? u) = ρ( x) f( u), x ∈ R N . No monotonicity condition is assumed upon f( u). Our method used to get the existence of the solution is based on sub-and supersolutions techniques. 相似文献
16.
In this paper, we study the existence of periodic solutions of Rayleigh equation
where f, g are continuous functions and p is a continuous and 2π-periodic function. We prove that the given equation has at least one 2π-periodic solution provided that f( x) is sublinear and the time map of equation x′′ + g( x) = 0 satisfies some nonresonant conditions. We also prove that this equation has at least one 2π-periodic solution provided that g( x) satisfies
and f( x) satisfies sgn( x)( f( x) − p( t)) ≥ c, for t ∈ R, | x| ≥ d with c, d being positive constants.Received: July 1, 2002; revised: February 19, 2003Research supported by the National Natural Science Foundation of China, No.10001025 and No.10471099, Natural Science Foundation of Beijing, No. 1022003 and by a postdoctoral Grant of University of Torino, Italy. 相似文献
17.
The paper introduces a notion of the Laplace operator of a polynomial p in noncommutative variables x = ( x
1,…, x
g
). The Laplacian Lap[ p, h] of p is a polynomial in x and in a noncommuting variable h. When all variables commute we have Lap[ p, h] = h
2Δ
x
p where Δ
x
p is the usual Laplacian. A symmetric polynomial in symmetric variables will be called harmonic if Lap[ p, h] = 0 and subharmonic if the polynomial q( x, h) := Lap[ p, h] takes positive semidefinite matrix values whenever matrices X
1,…, X
g
, H are substituted for the variables x
1,…, x
g
, h. In this paper we classify all homogeneous symmetric harmonic and subharmonic polynomials in two symmetric variables. We
find there are not many of them: for example, the span of all such subharmonics of any degree higher than 4 has dimension
2 (if odd degree) and 3 (if even degree). Hopefully, the approach here will suggest ways of defining and analyzing other partial
differential equations and inequalities.
Dedicated to Israel Gohberg on the occasion of his 80 th birthday.
All authors were partially supported by J.W. Helton’s grants from the NSF and the Ford Motor Co. and J. A. Hernandez was supported
by a McNair Fellowship. 相似文献
18.
We consider well‐posedness of the aggregation equation ∂ tu + div( uv) = 0, v = −▿ K * u with initial data in \input amssym ${\cal P}_2 {\rm (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ in dimensions 2 and higher. We consider radially symmetric kernels where the singularity at the origin is of order | x| α, α > 2 − d, and prove local well‐posedness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ for sufficiently large p < ps. In the special case of K( x) = | x|, the exponent ps = d/( d = 1) is sharp for local well‐posedness in that solutions can instantaneously concentrate mass for initial data in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ with p < ps. We also give an Osgood condition on the potential K( x) that guarantees global existence and uniqueness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ . © 2010 Wiley Periodicals, Inc. 相似文献
19.
We consider the Dirichlet problem for a class of quasilinear degenerate elliptic inclusions of the form ?div(𝒜( x, u, ? u)) + f( x) g( u) ∈ H( x, u, ? u), where 𝒜( x, u, ? u) is allowed to be degenerate. Without the general assumption that the multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz functions, we prove the existence of bounded solutions in weighed Sobolev space with the superlinear growth imposed on the nonlinearity g and the multifunction H( x, u, ? u) by using the Leray-Schauder fixed point theorem. Furthermore, we investigate the existence of extremal solutions and prove that they are dense in the solutions of the original system. Subsequently, a quasilinear degenerate elliptic control problem is considered and the existence theorem based on the proven results is obtained. 相似文献
20.
The predictive ratio is considered as a measure of spread for the predictive distribution. It is shown that, in the exponential families, ordering according to the predictive ratio is equivalent to ordering according to the posterior covariance matrix of the parameters. This result generalizes an inequality due to Chaloner and Duncan who consider the predictive ratio for a beta-binomial distribution and compare it with a predictive ratio for the binomial distribution with a degenerate prior. The predictive ratio at x1 and x2 is defined to be pg( x1) pg( x2)/[ pg(
)] 2 = hg( x1, x2), where pg( x1) = ∫ ƒ( x1θ) g(θ) dθ is the predictive distribution of x1 with respect to the prior g. We prove that hg( x1, x2) ≥ hg*( x1, x2) for all x1 and x2 if ƒ( xθ) is in the natural exponential family and Cov gx(θ) ≥ Cov g*x(θ) in the Loewner sense, for all x on a straight line from x1 to x2. We then restrict the class of prior distributions to the conjugate class and ask whether the posterior covariance inequality obtains if g and g* differ in that the “sample size” 相似文献
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