共查询到20条相似文献,搜索用时 843 毫秒
1.
M. Langenbruch 《manuscripta mathematica》2000,103(2):241-263
Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ
n
. Let L(P
m
) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P
m
. Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ
n
×(ℝ\{ 0}) for any Q∈L(P
m
) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P
m
must be locally hyperbolic.
Received: 24 January 2000 相似文献
2.
Byron L. Walden 《Journal d'Analyse Mathématique》1994,63(1):231-253
In 1981, Hayman and Wu proved that for any simply connected domain Ω and any Riemann mappingF: Ω →D,F′ ∈ L1 (L ∩ Ω), whereL is any line in the complex plane. Several years later, Fernández, Heinonen and Martio showed that there is anε > 0 such thatF′ ∈ L1+∈(L ∩ Ω). The question arises as to which curves other than lines satisfy such a statement. A curve Γ is said to be Ahlfors-David
regular if there is a constantA such that for any B(x, r) (the disk of radiusr centered atx), l(Γ ∩ B(x, r))≤ Ar. The major result of the paper is the following theorem: Let Γ be an Ahlfors-David regular curve with constantA. Then there exists an∈ > 0, depending only onA, such thatF′ ∈ L1+∈(Γ ∩ Ω). This result is the synthesis of the extension of Fernández, Heinonen and Martio, and the result of Bishop and Jones
showing thatF′ ∈ L1(Γ ∩ Ω). The proof of the results uses a stopping-time argument which seeks out places in the curve where small pieces may
be added in order to control the portions of the curve where |F′ | is large. This is accomplished with an estimate on the
vanishing of the harmonic measure of the curve in such places. The paper also includes simpler arguments for the special cases
where Γ = ∂Ω and Γ ⊂Ω. 相似文献
3.
In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” uf (·, T) is studied, where uf (x,t) is the solution of the initial boundary-value problem utt−Δu=0 in Ω×(0,T), u|t<0=0, u|∂Ω×(0,T)=f, and the (singular) control f runs over the class L2((0,T); H−m (∂Ω)) (m>0). The following result is established. Let ΩT={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝn (diam Ω<∞) filled with waves by a final instant of time t=T, let T*=inf{T : ΩT=Ω} be the time of filling the whole domain Ω. We introduce the notation Dm=Dom((−Δ)m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H2(Ω)∩H
0
1
(Ω);D−m=(Dm)′;D−m(ΩT)={y∈D−m:supp y ⋐ ΩT. If T<T., then the reachable set R
m
T
={ut(·, T): f ∈ L2((0,T), H−m (∂Ω))} (∀m>0), which is dense in D−m(ΩT), does not contain the class C
0
∞
(ΩT). Examples of a ∈ C
0
∞
, a ∈ R
m
T
, are presented.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21.
Translated by T. N. Surkova. 相似文献
4.
Andreas Fr?hlich 《Annali dell'Universita di Ferrara》2000,46(1):11-19
We consider weights of Muckenhoupt classA
q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝn we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve
the weak Neumann problem for the Laplace equation in weighted spaces on ℝn, ℝn
+, on bounded and on exterior domains Ω with boundary of classC
1, which will yield the Helmholtz decomposition ofL
ω
q(Ω)n for general ω∈A
q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of
Farwig and Sohr [2] where the Helmholtz decomposition ofL
ω
p(Ω)n is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood
of ϖΩ.
Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝn, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝn, ℝn + in domini limitati ed in domini esterni con frontiera di classeC 1, che conduce alla decomposizione di Helmholtz diL ω q(Ω)n per un qualsiasi ω∈A q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL ω q(Ω)n è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.相似文献
5.
Aissa Guesmia 《Israel Journal of Mathematics》2001,125(1):83-92
We consider in this paper the evolution systemy″−Ay=0, whereA =∂
i(aij∂j) anda
ij ∈C
1 (ℝ+;W
1,∞ (Ω)) ∩W
1,∞ (Ω × ℝ+), with initial data given by (y
0,y
1) ∈L
2(Ω) ×H
−1 (Ω) and the nonhomogeneous conditiony=v on Γ ×]0,T[. Exact controllability means that there exist a timeT>0 and a controlv such thaty(T, v)=y′(T, v)=0. The main result of this paper is to prove that the above system is exactly controllable whenT is “sufficiently large”. Moreover, we obtain sharper estimates onT. 相似文献
6.
In this paper we consider positive semigroups on Lp(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups
as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L∞(Ω) is real and satisfies b ≥ δ for some δ > 0.
In memoriam Helmut H. Schaefer 相似文献
7.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
8.
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
9.
Zhi-jian QIU Department of Economic Mathematics Southwestern University of Finance Economics Chengdu China 《中国科学A辑(英文版)》2007,50(3):305-312
For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk. 相似文献
10.
For an analytic function f on the hyperbolic domain Ω inC, the following conclusions are obtained: (i) f∈B(Ω)=BMO A(Ω,m) if and only ifRef∈Bh(Ω)=BMOH(Ω,m). (ii) QBh(Ω)=Bh(Ω)(BMOH
n(Ω,m)=BMOH(Ω,m)) if and only ifC(Ω)=inf{λΩ(z)·δΩ(z):z∈Ω}>0. Also, some applications to automorphic functions are considered.
This research was supported by the Doctoral Program Foundation of Institute of Higher Education. 相似文献
11.
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
12.
One proves that Ay=aΔ y+b⋅ \nabla y , D(A)={y∈ H
1
(Ω ); aΔ y+b⋅ \nabla y∈ H
0
1
(Ω ), \sqrt a Δ y∈ L
2
(Ω )} generates, under suitable conditions on a and b , a C
0
-analytic semigroup on H
1
(Ω) .
September 5, 1999 相似文献
13.
Let T and S be invertible measure preserving transformations of a probability measure space (X, ℬ, μ). We prove that if the group generated by T and S is nilpotent, then exists in L
2-norm for any u, v∈L
∞(X, ℬ, μ). We also show that for A∈ℬ with μ(A)>0 one has . By the way of contrast, we bring examples showing that if measure preserving transformations T, S generate a solvable group, then (i) the above limits do not have to exist; (ii) the double recurrence property fails, that
is, for some A∈ℬ, μ(A)>0, one may have μ(A∩T
-n
A∩S
-
n
A)=0 for all n∈ℕ. Finally, we show that when T and S generate a nilpotent group of class ≤c, in L
2(X) for all u, v∈L
∞(X) if and only if T×S is ergodic on X×X and the group generated by T
-1
S, T
-2
S
2,..., T
-c
S
c
acts ergodically on X.
Oblatum 19-V-2000 & 5-VII-2001?Published online: 12 October 2001 相似文献
14.
S. R. Hayrapetyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(4):221-238
The paper considers a class of regular, hypoelliptic in x
1, two-dimensional operators P(D) = P(D
1,D
2) in rather wide strip Ω
H
= {x = (x
1; x
2) ∈ $
\mathbb{E}
$
\mathbb{E}
2, |x
1| < H, x
2 ∈ $
\mathbb{E}
$
\mathbb{E}
1}. It is proved the infinite differentiability in Ω
H
of those generalized solutions of the equation P(D)
u
= 0, for which D
2
j
u ∈ L
2(Ω
H
), j = 0, …, ord
x2
P. 相似文献
15.
In this paper we apply Bishop-Phelps property to show that if X is a Banach space and G X is the maximal subspace so that G⊥ = {x* ∈ X*|x*(y) = 0; y∈ G} is an L-summand in X*, then L1(Ω,G) is contained in a maximal proximinal subspace of L1(Ω,X). 相似文献
16.
The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density.
The Neumann problem is examined in a bounded n-dimensional domain Ω+ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω− = R
n
\Ω+. Let Γ be the common boundary of the domains Ω±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R
n
, and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω− with a cusp of an inward peak may be represented as Vρ−, where ρ− ∈ Tr(Γ)* is uniquely determined for all Ψ− ∈ Tr(Γ)*. If Ω+ is a domain with an inward peak and if Ψ+ ∈ Tr(Γ)*, Ψ+ ⊥ 1, then the solution of the Neumann problem for Ω+ has the representation u
+ = Vρ+ for some ρ+ ∈ Tr(Γ)* which is unique up to an additive constant ρ0, ρ0 = V
−1(1). These results do not hold for domains with outward peak. 相似文献
17.
Jack D. Koronel 《Israel Journal of Mathematics》1976,24(2):119-138
The paper gives a necessary and sufficient condition for the embedding of the Orlicz-Sobolev spaceW
kLA (Ω) inC(Ω). The same condition is also found to be necessary and sufficient so that a continuous function inW
kLA (Ω) be differentiable of orderk almost everywhere in Ω. 相似文献
18.
SoitM(Ω, η, ξ,g) une variété à (2m+1)-dimensions presque cosymplectique (i. e. Ω∈Λ2
M est de rang 2m et Ω
m
Λη≠0). On définitM comme étant une variété semi-cosymplectique si en termes ded
ω-cohomologie la paire (Ω, η) satisfait àdη=0,d
−cη Ω=Ψ∈Λ3
M,c=constant. Dans ce cas le champ vectoriel de structure ξ=b
−1(η) est un champ conforme horizontal et siM est une forme-espace elle est nécessairement du type hyperbolique. Différentes propriétés de cette structure sont étudiés
et le cas oùM est une variété para Sasakienne dans le sens large est discuté. 相似文献
19.
Guido Cortesani 《Annali dell'Universita di Ferrara》1997,43(1):27-49
Let Ω be an open and bounded subset ofR
n
with locally Lipschitz boundary. We prove that the functionsv∈SBV(Ω,R
m
) whose jump setS
vis essentially closed and polyhedral and which are of classW
k, ∞ (S
v,R
m) for every integerk are strongly dense inGSBV
p(Ω,R
m
), in the sense that every functionu inGSBV
p(Ω,R
m
) is approximated inL
p(Ω,R
m
) by a sequence of functions {v
k{j∈N with the described regularity such that the approximate gradients ∇v
jconverge inL
p(Ω,R
nm
) to the approximate gradient ∇u and the (n−1)-dimensional measure of the jump setsS
v
j converges to the (n−1)-dimensional measure ofS
u. The structure ofS
v can be further improved in casep≤2.
Sunto Sia Ω un aperto limitato diR n con frontiera localmente Lipschitziana. In questo lavoro si dimostra che le funzioniv∈SBV(Ω,R m ) con insieme di saltoS v essenzialmente chiuso e poliedrale che sono di classeW k, ∞ (S v,R m ) per ogni interok sono fortemente dense inGSBV p(Ω,R m ), nel senso che ogni funzioneu∈GSBV p(Ω,R m ) è approssimata inL p(Ω,R m ) da una successione di funzioni {v j}j∈N con la regolaritá descritta tali che i gradienti approssimati ∇v jconvergono inL p(Ω,R nm ) al gradiente approssimato ∇u e la misura (n−1)-dimensionale degli insiemi di saltoS v jconverge alla misura (n−1)-dimensionale diS u. La struttura diS vpuó essere migliorata nel caso in cuip≤2.相似文献
20.
Let Ω be a domain with piecewise smooth boundary. In general, it is impossible to obtain a generalized solution u ∈ W
2
2
(Ω) of the equation Δ
x
2
u = f with the boundary conditions u = Δxu = 0 by solving iteratively a system of two Poisson equations under homogeneous Dirichlet conditions. Such a system is obtained
by setting v = −Δu. In the two-dimensional case, this fact is known as the Sapongyan paradox in the theory of simply supported
polygonal plates. In the present paper, the three-dimensional problem is investigated for a domain with a smooth edge Γ. If
the variable opening angle α ∈ C∞(Γ) is less than π everywhere on the edge, then the boundary-value problem for the biharmonic equation is equivalent to the
iterated Dirichlet problem, and its solution u inherits the positivity preserving property from these problems. In the case
α ∈ (π 2π), the procedure of solving the two Dirichlet problems must be modified by permitting infinite-dimensional kernel
and co-kernel of the operators and determining the solution u ∈ W
2
2
(Ω) by inverting a certain integral operator on the contour Γ. If α(s) ∈ (3π/2,2π) for a point s ∈ Γ, then there exists a
nonnegative function f ∈ L2(Ω) for which the solution u changes sign inside the domain Ω. In the case of crack (α = 2π everywhere on Γ), one needs to
introduce a special scale of weighted function spaces. In this case, the positivity preserving property fails. In some geometrical
situations, the problems on well-posedness for the boundary-value problem for the biharmonic equation and the positivity property
remain open. Bibliography: 46 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 153–198. 相似文献