共查询到20条相似文献,搜索用时 171 毫秒
1.
Maria Manfredini 《Annali di Matematica Pura ed Applicata》2009,188(3):417-428
In this paper we are concerned with a family of elliptic operators represented as sum of square vector fields: in , where Δ is the Laplace operator, m < n, and the limit operator is hypoelliptic. Here we establish Schauder’s estimates, uniform with respect to the parameter ϵ, of solution of the approximated
equation L
ϵ
u = f, using a modification of the lifting technique of Rothschild and Stein. These estimates can be used in particular while studying
regularity of viscosity solutions of nonlinear equations represented in terms of vector fields.
相似文献
2.
Second-order half-linear differential equation (H): on the finite interval I = (0,1] will be studied, where , p > 1 and the coefficient f(x) > 0 on I, , and . In case when p = 2, the equation (H) reduces to the harmonic oscillator equation (P): y′′ + f(x)y = 0. In this paper, we study the oscillations of solutions of (H) with special attention to some geometric and fractal properties of the graph . We establish integral criteria necessary and sufficient for oscillatory solutions with graphs having finite and infinite
arclength. In case when , λ > 0, α > p, we also determine the fractal dimension of the graph G(y) of the solution y(x). Finally, we study the L
p
nonintegrability of the derivative of all solutions of the equation (H).
相似文献
3.
Xavier Cabré Antonio Capella Manel Sanchón 《Calculus of Variations and Partial Differential Equations》2009,34(4):475-494
We consider semi-stable, radially symmetric, and decreasing solutions of − Δ
p
u = g(u) in the unit ball of , where p > 1, Δ
p
is the p-Laplace operator, and g is a locally Lipschitz function. For this class of radial solutions, which includes local minimizers, we establish pointwise,
L
q
, and W
1,q
estimates which are optimal and do not depend on the specific nonlinearity g. Among other results, we prove that every radially decreasing and semi-stable solution u belonging to W
1,p
(B
1) is bounded whenever n < p + 4p/(p − 1). Under standard assumptions on the nonlinearity g(u) = λf (u), where λ > 0 is a parameter, it is proved that the corresponding extremal solution u
* is semi-stable, and hence, it enjoys the regularity stated in our main result. 相似文献
4.
We consider the generalized Gagliardo–Nirenberg inequality in in the homogeneous Sobolev space with the critical differential order s = n/r, which describes the embedding such as for all q with p ≦ q < ∞, where 1 < p < ∞ and 1 < r < ∞. We establish the optimal growth rate as q → ∞ of this embedding constant. In particular, we realize the limiting end-point r = ∞ as the space of BMO in such a way that with the constant C
n
depending only on n. As an application, we make it clear that the well known John–Nirenberg inequality is a consequence of our estimate. Furthermore,
it is clarified that the L
∞-bound is established by means of the BMO-norm and the logarithm of the -norm with s > n/r, which may be regarded as a generalization of the Brezis–Gallouet–Wainger inequality. 相似文献
5.
Piotr Kot 《Czechoslovak Mathematical Journal》2009,59(2):371-379
We solve the following Dirichlet problem on the bounded balanced domain with some additional properties: For p > 0 and a positive lower semi-continuous function u on ∂Ω with u(z) = u(λ z) for |λ| = 1, z ∈ ∂Ω we construct a holomorphic function f ∈ (Ω) such that for z ∈ ∂Ω, where = {λ ∈ ℂ: |λ| < 1}.
相似文献
6.
Thomas Bartsch Zhi-Qiang Wang Juncheng Wei 《Journal of Fixed Point Theory and Applications》2007,2(2):353-367
We consider the existence of bound states for the coupled elliptic system
where n ≤ 3. Using the fixed point index in cones we prove the existence of a five-dimensional continuum of solutions (λ1, λ2, μ
1, μ
2, β, u
1, u
2) bifurcating from the set of semipositive solutions (where u
1 = 0 or u
2 = 0) and investigate the parameter range covered by .
Dedicated to Albrecht Dold and Edward Fadell 相似文献
7.
P. Quittner W. Reichel 《Calculus of Variations and Partial Differential Equations》2008,32(4):429-452
Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂ν
u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L
∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s|
p
) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that
p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) = s
p
then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of
∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential
equation is of the form h(x, u) with h satisfying suitable growth conditions. 相似文献
8.
Thomas Westerbäck 《Designs, Codes and Cryptography》2007,42(3):335-355
A maximal partial Hamming packing of is a family of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in . The number of translates of Hamming codes in is the packing number, and a partial Hamming packing is strictly partial if the family does not constitute a partition of .
A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This
condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that
the packing number p, for any maximal strictly partial Hamming packing of , n = 2
m
−1, satisfies .
It is also proved that for any n equal to 2
m
−1, , there exist maximal strictly partial Hamming packings of with packing numbers n−10,n−9,n−8,...,n−1. This implies that the upper bound is tight for any n = 2
m
−1, .
All packing numbers for maximal strictly partial Hamming packings of , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5,6,7,...,13 and 14.
相似文献
9.
Wolfgang Reichel 《Annali di Matematica Pura ed Applicata》2009,188(2):235-245
For a bounded convex domain and consider the unit- density Riesz-potential . We show in this paper that u = const. on ∂G if and only if G is a ball. This result corresponds to a theorem of L.E. Fraenkel, where the ball is characterized by the Newtonian-potential
(α = 2) of unit density being constant on ∂G. In the case α = N the kernel |x − y|
α-N
is replaced by − log|x − y| and a similar characterization of balls is given. The proof relies on a recent variant of the moving plane method which
is suitable for Green-function representations of solutions of (pseudo-)differential equations of higher-order.
相似文献
10.
Luciana Angiuli Michele MirandaJr Diego Pallara Fabio Paronetto 《Annali di Matematica Pura ed Applicata》2009,188(2):297-331
Given a uniformly elliptic second order operator on a possibly unbounded domain , let (T(t))
t≥0 be the semigroup generated by in L
1(Ω), under homogeneous co-normal boundary conditions on ∂Ω. We show that the limit as t → 0 of the L
1-norm of the spatial gradient D
x
T(t)u
0 tends to the total variation of the initial datum u
0, and in particular is finite if and only if u
0 belongs to BV(Ω). This result is true also for weighted BV spaces. A further characterization of BV functions in terms of the short-time behaviour of (T(t))
t≥0 is also given.
相似文献
11.
Flávio Dickstein Miguel Loayza 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(1):1-23
We consider the Cauchy problem for the weakly coupled parabolic system ∂
t
w
λ−Δ w
λ = F(w
λ) in R
N
, where λ > 0, w
λ = (u
λ, v
λ), F(w
λ) = (v
λ
p
, u
λ
q
) for some p, q ≥ 1, pq > 1, and , for some nonnegative functions φ1, φ2
C
0(R
N
). If (p, q) is sub-critical or either φ1 or φ2 has slow decay at ∞, w
λ blows up for all λ > 0. Under these conditions, we study the blowup of w
λ for λ small.
相似文献
12.
Filippo Gazzola Hans-Christoph Grunau 《Calculus of Variations and Partial Differential Equations》2007,30(3):389-415
We are interested in stability/instability of the zero steady state of the superlinear parabolic equation u
t
+ Δ2
u = |u|
p-1
u in , where the exponent is considered in the “super-Fujita” range p > 1 + 4/n. We determine the corresponding limiting growth at infinity for the initial data giving rise to global bounded solutions.
In the supercritical case p > (n + 4)/(n−4) this is related to the asymptotic behaviour of positive steady states, which the authors have recently studied. Moreover,
it is shown that the solutions found for the parabolic problem decay to 0 at rate t
−1/(p-1). 相似文献
13.
Curve shortening in a Riemannian manifold 总被引:1,自引:0,他引:1
In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior
of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and
, then for every n > 0,
. Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S
1, if γ0 is a ramp, then we have a global flow which converges to a closed geodesic in C
∞ norm. As an application, we prove the theorem of Lyusternik and Fet.
相似文献
14.
By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation
given by ( a1, …, an). Assume that either a1, …, an are positive rational numbers or
for some
Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝn must be infinite 相似文献
15.
Sandra Lucente 《Annali dell'Universita di Ferrara》2006,52(2):317-335
Abstract In this paper, we deal with some global existence results for the large data smooth solutions of the Cauchy Problem associated
with the semilinear weakly hyperbolic equations
Here u=u(x,t),
and for λ≥ 0, aλ≥ 0 is a continuous function that behaves as |t–t0|λ close to some t0>0. We conjecture the existence of a critical exponent pc(λ1,λ2,n) such that for p≤ pc(λ1,λ2,n) a global existence theorem holds. For suitable λ1,λ2,n, we recall some known results and add new ones.
Keywords: Critical exponents for semilinear equations, Weak hyperbolicity 相似文献
16.
Let be a C
2 map and let Spec(Y) denote the set of eigenvalues of the derivative DY
p
, when p varies in . We begin proving that if, for some ϵ > 0, then the foliation with made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case
of Jelonek’s Jacobian Conjecture for polynomial maps of
The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil
Grant 03/03107-9. 相似文献
17.
De-xiang Ma Wei-gao Ge Xue-gang Chen 《应用数学学报(英文版)》2005,21(4):661-670
In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{( φp(u'))'+q(t)f(t,u,u')=0,0〈t〈1,u(0)=∑i=1^nαiu(ξi),u'(1)=∑i=1^nβiu'(ξi),whereφp(s)=|s|^p-2s,p≥2;ξi∈(0,1)(i=1,2,…,n),0≤αi,βi〈1(i=1,2,…n),0≤∑i=1^nαi,∑i=1^nβi〈1,and q(t) may be singular at t=0,1,f(t,u,u')may be singular at u'=0 相似文献
18.
Tetsutaro Shibata 《Annales Henri Poincare》2008,9(6):1217-1227
We consider the nonlinear eigenvalue problem
,
where f(u) = u
p
+ h(u) (p > 1) and λ > 0 is a parameter. Typical example of h(u) is with 1 < q < (p+ 1)/2. We establish the precise asymptotic formula for L
m
-bifurcation branch λ = λ
m
(α) of positive solutions as α → ∞, where α > 0 is the L
m
-norm of the positive solution associated with .
Submitted: September 27, 2007. Accepted: May 28, 2008. 相似文献
19.
Piotr Niemiec 《Rendiconti del Circolo Matematico di Palermo》2008,57(3):391-399
The aim of the paper is to prove that every f ∈ L
1([0,1]) is of the form f = , where j
n,k
is the characteristic function of the interval [k- 1 / 2
n
, k / 2
n
) and Σ
n=0∞Σ
k=12n
|a
n,k
| is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b
n,k
)
n≧0
k=1,...,2n
of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).
相似文献
20.
Andrey Shishkov Laurent Véron 《Calculus of Variations and Partial Differential Equations》2008,33(3):343-375
We study the limit behaviour of solutions of with initial data k
δ
0 when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r
β
, β > N(p − 1) − 2, we prove that the limit function u
∞ is an explicit very singular solution, while such a solution does not exist if β ≤ N(p − 1) − 2. If lim
inf
r→ 0
r
2 ln (1/h(r)) > 0, u
∞ has a persistent singularity at (0, t) (t ≥ 0). If , u
∞ has a pointwise singularity localized at (0, 0). 相似文献