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1.
Carleman estimates for one-dimensional degenerate heat equations 总被引:1,自引:0,他引:1
In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the
space domain.
We derive new Carleman estimates for the degenerate parabolic equation
$$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1}
\right), $$ where the function a mainly satisfies
$$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt
0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested
in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=xα (1 − x)β with α, β ∈ [0, 2).
As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation
$$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$
The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=xα with α ∈[0, 2).
Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday 相似文献
2.
In this paper, we prove higher integrability results for the gradient of the solutions of some elliptic equations with degenerate
coercivity whose prototype is
where for example, a(x,u)=(1+|u|)−θ with θ ∈ (0,1). We study the same problem for minima of functionals closely related to the previous equation. 相似文献
3.
We exhibit, for any integerg≥2, an infinite sequenceA ∈B
2[g] such that
. Furthermore, we obtain better estimates for small values ofg. For instance, we exhibit an infinite sequenceA ∈B
2[2] such that
Partially supported by Colciencias, Colombia and Universidad del Cauca. 相似文献
4.
Nataliya V. Smorodina 《Acta Appl Math》2007,97(1-3):239-250
Let ξ(t),t∈[0,1] be a strictly stable Lévy process with the index of stability α∈(0,2). By ℘
ξ
we denote the law of ξ in the Skorokhod space
. For arbitrary ξ we construct ℘
ξ
-quasi-invariant semigroup of transformations of
. Under some nondegeneracy condition on the spectral measure of the stable process we construct ℘
ξ
-quasi-invariant group of transformations of
. In symmetric case this group is a group of the invariant transformations.
相似文献
5.
Cao Jiading 《分析论及其应用》1989,5(2):99-109
Let an≥0 and F(u)∈C [0,1], Sikkema constructed polynomials:
, ifα
n
≡0, then Bn (0, F, x) are Bernstein polynomials.
Let
, we constructe new polynomials in this paper:
Q
n
(k)
(α
n
,f(t))=d
k
/dx
k
B
n+k
(α
n
,F
k
(u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα
n
≡0, k=1, then Qn
(1) (0, f(t), x) are Kantorovič polynomials Pn(f). Ifα
n
=0, k=2, then Qn
(2), (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is:
Theorem 2. Let 1≤p≤∞, in order that for every f∈LP [0, 1],
, it is sufficient and necessary that
,
§ 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define[1–2],[8–10]:
.
As usual, for the space Lp [a,b](1≤p<∞), we have
and L[a, b]=l1[a, b].
Letα
n
⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials
[3] [4].
The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports. 相似文献
6.
Consider Hill’s operator
in which the potential q(x) is an almost surely continuous and rotation invariant Gaussian process on the circle x∈[0,1). Viewing the classical Riccati map as a change of measure, we establish functional integral formulas for the probability
density function of the ground state energy and also determine the density’s shape. 相似文献
7.
Mario Petrich 《Czechoslovak Mathematical Journal》2006,56(1):27-46
Let S be a regular semigroup and E(S) be the set of its idempotents. We call the sets S(e, f)f and eS(e, f) one-sided sandwich sets and characterize them abstractly where e, f ∈ E(S). For a, a′ ∈ S such that a = aa′a, a′ = a′aa′, we call S(a) = S(a′a, aa′) the sandwich set of a. We characterize regular semigroups S in which all S(e; f) (or all S(a)) are right zero semigroups (respectively are trivial) in several ways including weak versions of compatibility of the natural
order.
For every a ∈ S, we also define E(a) as the set of all idempotets e such that, for any congruence ϱ on S, aϱa
2 implies that aϱe. We study the restrictions on S in order that S(a) or
be trivial. For
, we define
on S by a
b if
. We establish for which S are
or
congruences. 相似文献
8.
We prove a functional large deviations principle for the family of random functions
, where { Zt, t ∈, [0, 1]} is a real-valued centered Gaussian process. Bibliography: 19 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 169–181. 相似文献
9.
Riccardo De Arcangelis 《Annali dell'Universita di Ferrara》1989,35(1):135-145
Summary Letf: (x, z)∈R
n×Rn→f(x, z)∈[0, +∞] be measurable inx and convex inz.
It is proved, by an example, that even iff verifies a condition as|z|
p≤f(x, z)≤Λ(a(x)+|z|q) with 1<p<q,a∈L
loc
s
(R
n),s>1, the functional
that isL
1(Ω)-lower semicontinuous onW
1,1(Ω), does not agree onW
1,1(Ω) with its relaxed functional in the topologyL
1(Ω) given by inf
Riassunto Siaf: (x, z)∈R n×Rn→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| p≤f(x, z)≤Λ(a(x)+|z|q) con 1<p<q,a∈L loc s (R n),s>1, il funzionale , che èL 1(Ω)-semicontinuo inferiormente suW 1,1(Ω), non coincide suW 1,1(Ω) con il suo funzionale rilassato nella topologiaL 1(Ω) definito da inf相似文献
10.
K. J. Wirths 《分析论及其应用》1996,12(3):98-100
Let
be such that |p(eiq)|≤1 for ϕ∈R and |p(1)|=a∈[0,1]. An inequality of Dewan and Govil for the sum |av|+|an|, 0≤u<v≤n is sharpened. 相似文献
11.
If f∈Lp[0, 1], let fp be its best Lp-approximant by convex functions. It is shown that if
exists uniformly on closed subintervals of (0,1).
This research was partially supported by Grant No. 020-033-58 from the Faculty Research Committee, Idaho State University. 相似文献
12.
E. A. Gorin 《Functional Analysis and Its Applications》2011,45(1):73-76
Let X be an Abelian semigroup such that the following conditions hold: (i) if x × y = II (II is the identity element), then x = y = II; (ii) the set {{x, y}: x × y = a} is finite for any a ∈ X. Let Λ be any field, and let ℰ be the algebra of all Λ-valued functions on X. The convolution of u, υ ∈ ℰ is defined by
)( x ) = u( a )v( b ):a b = x . |
#xA;
\left( {u*v} \right)\left( x \right) = \sum {\left\{ {u\left( a \right)v\left( b \right):a \times b = x} \right\}.}
相似文献
13.
Uri Fixman 《Integral Equations and Operator Theory》2000,37(1):9-19
LetA be the linear operator inL
p
(0, 1), 1<p<∞,p≠2, defined by
,x∈L
p
(0, 1),s∈[0,1]. We show that the real values of numbers in the numerical range ofA have maximum
, whereq=p/(p−1). This amounts to an inequality between integrals, for which we determine the case of equality. 相似文献
14.
Katalin Gyarmati 《The Ramanujan Journal》2008,17(3):387-403
Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let
15.
Liu Chuan ZENG 《数学学报(英文版)》2006,22(2):407-416
Let G be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banaeh space E with a Frechet differentiable norm, and T = {Tt : t ∈ G} be a continuous representation of G as nearly asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F(T) of T in C is nonempty. It is shown that if G is right reversible, then for each almost-orbit u(.) of T, ∩s∈G ^-CO{u(t) : t ≥ s} ∩ F(T) consists of at most one point. Furthermore, ∩s∈G ^-CO{Ttx : t ≥ s} ∩ F(T) is nonempty for each x ∈ C if and only if there exists a nonlinear ergodic retraction P of C onto F(T) such that PTs - TsP = P for all s ∈ G and Px ∈^-CO{Ttx : s ∈ G} for each x ∈ C. This result is applied to study the problem of weak convergence of the net {u(t) : t ∈ G} to a common fixed point of T. 相似文献
16.
17.
On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives
For 2-periodic functions
and arbitrary q [1, ] and p (0, ], we obtain the new exact Kolmogorov-type inequality
which takes into account the number of changes in the sign of the derivatives (x
(k)) over the period. Here, = (r – k + 1/q)/(r + 1/p),
r
is the Euler perfect spline of degree r,
and
. The inequality indicated turns into the equality for functions of the form x(t) = a
r
(nt + b), a, b R, n N. We also obtain an analog of this inequality in the case where k = 0 and q = and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines. 相似文献
18.
We propose a class of self-adaptive proximal point methods suitable for degenerate optimization problems where multiple minimizers
may exist, or where the Hessian may be singular at a local minimizer. If the proximal regularization parameter has the form
where η∈[0,2) and β>0 is a constant, we obtain convergence to the set of minimizers that is linear for η=0 and β sufficiently small, superlinear for η∈(0,1), and at least quadratic for η∈[1,2). Two different acceptance criteria for an approximate solution to the proximal problem are analyzed. These criteria
are expressed in terms of the gradient of the proximal function, the gradient of the original function, and the iteration
difference. With either acceptance criterion, the convergence results are analogous to those of the exact iterates. Preliminary
numerical results are presented using some ill-conditioned CUTE test problems.
This material is based upon work supported by the National Science Foundation under Grant Nos. 0203270, 0619080, and 0620286. 相似文献
19.
X(t) (t∉[0,∞)) is a subordinator with its upper index β less than one, g(u) is the index function ofX(t), andX(t), andX[0,1]={xϕR:X(t)=x} for sometϕ[0,1]{. If φ(s)(sϕ(0,1)) is a measure function andh
, then
20.
Ilya A. Krishtal Benjamin D. Robinson Guido L. Weiss Edward N. Wilson 《Journal of Geometric Analysis》2007,17(1):87-96
An orthonormal wavelet system in ℝd, d ∈ ℕ, is a countable collection of functions {ψ
j,k
ℓ
}, j ∈ ℤ, k ∈ ℤd, ℓ = 1,..., L, of the form
that is an orthonormal basis for L2 (ℝd), where a ∈ GLd (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L
= d = 1, ψ1(x) = ψ(x) = κ[0,1/2)(x) − κ[l/2,1)
(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate
products Φ(x1, x2, ..., xd) = φ1 (x1)φ2(x2) ... φd(xd) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to
find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems.
For example, if a = (
1-1
1 1
) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling
functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct
considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed
in [7] and involve dilations by matrices that are products of the form ajb, j ∈ ℤ, where a ∈ GLd(ℝ) has some “expanding” property and b belongs to a group of matrices in GLd(ℝ) having |det b| = 1. 相似文献
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