共查询到20条相似文献,搜索用时 31 毫秒
1.
David Kalaj 《Monatshefte für Mathematik》2012,36(1):205-229
The conformal deformations are contained in two classes of mappings quasiconformal and harmonic mappings. In this paper we consider the intersection of these classes. We show that, every K quasiconformal harmonic mapping between surfaces with boundary is a Lipschitz mapping. This extends some recent results of several authors where the same problem has been considered for plane domains. As an application it is given an explicit Lipschitz constant of normalized isothermal coordinates of a disk-type minimal surface in terms of boundary curve only. It seems that this kind of estimates are new for conformal mappings of the unit disk onto a Jordan domain as well. 相似文献
2.
In the paper, a global optimization problem is considered where the objective function f (x) is univariate, black-box, and its first derivative f ′(x) satisfies the Lipschitz condition with an unknown Lipschitz constant K. In the literature, there exist methods solving this problem by using an a priori given estimate of K, its adaptive estimates, and adaptive estimates of local Lipschitz constants. Algorithms working with a number of Lipschitz
constants for f ′(x) chosen from a set of possible values are not known in spite of the fact that a method working in this way with Lipschitz
objective functions, DIRECT, has been proposed in 1993. A new geometric method evolving its ideas to the case of the objective
function having a Lipschitz derivative is introduced and studied in this paper. Numerical experiments executed on a number
of test functions show that the usage of derivatives allows one to obtain, as it is expected, an acceleration in comparison
with the DIRECT algorithm.
This research was supported by the RFBR grant 07-01-00467-a and the grant 4694.2008.9 for supporting the leading research
groups awarded by the President of the Russian Federation. 相似文献
3.
We consider two conformally invariant metrics in proper subdomains of euclideann-spaceR
n. We show that Lipschitz mappings in these metrics include the class of quasiconformal mappings as a proper subclass, yet
these Lipschitz mappings have many properties similar to those of quasiconformal mappings.
Research supported in part by the U.S. National Science Foundation and the A.P. Sloan Foundation.
Research supported in part by the Alexander von Humboldt Foundation. 相似文献
4.
5.
Roland Lemmert 《Applicable analysis》2013,92(1-3):65-73
The shooting method is used to prove existence and uniqueness of solutions to (1) under various types of Lipschitz conditions on f. Since the shooting function turns out to be strictly monotone, a priori estimates for the solutions are easily obtained. 相似文献
6.
Let f be an orientation-preserving circle homeomorphism and Φ the Douady-Earle extension of f. In this paper, we show that the quasiconformality and asymptotic conformality of Φ are local properties; i.e., if f is quasisymmetric or symmetric on an arc of the unit circle, then Φ is quasiconformal or asymptotically conformal nearby.
Furthermore, our methods enable us to conclude the global quasiconformality and asymptotic conformality from local properties.
In the quasiconformal case, our methods also enable us to provide an upper bound for the maximal dilatation of Φ on a neighborhood
of the arc in the open unit disk in terms of the cross-ratio distortion norm of f on the arc. 相似文献
7.
The rate of convergence of q-Bernstein polynomials for 总被引:3,自引:0,他引:3
In the note, we obtain the estimates for the rate of convergence for a sequence of q-Bernstein polynomials {Bn,q(f)} for 0<q<1 by the modulus of continuity of f, and the estimates are sharp with respect to the order for Lipschitz continuous functions. We also get the exact orders of convergence for a family of functions , and the orders do not depend on α, unlike the classical case. 相似文献
8.
Letf(z, t) be a subordination chain fort ∈ [0, α], α>0, on the Euclidean unit ballB inC
n. Assume thatf(z) =f(z, 0) is quasiconformal. In this paper, we give a sufficient condition forf to be extendible to a quasiconformal homeomorphism on a neighbourhood of
. We also show that, under this condition,f can be extended to a quasiconformal homeomorphism of
onto itself and give some applications.
Partially supported by Grant-in-Aid for Scientific Research (C) no. 14540195 from Japan Society for the Promotion of Science,
2004. 相似文献
9.
Mihai Cristea 《Complex Analysis and Operator Theory》2011,5(3):863-880
The paper continues the work of Royster (Duke Math J 19:447–457, 1952), Mocanu [Mathematica (Cluj) 22(1):77–83, 1980; Mathematica
(Cluj) 29:49–55, 1987], Cristea [Mathematica (Cluj) 36(2):137–144, 1994; Complex Var 42:333–345, 2000; Mathematica (Cluj)
43(1):23–34, 2001; Mathematica (Cluj), 2010, to appear; Teoria Topologica a Functiilor Analitice, Editura Universitatii Bucuresti,
Romania, 1999] of extending univalence criteria for complex mappings to C
1 mappings. We improve now the method of Loewner chains which is usually used in complex univalence theory for proving univalence
criteria or for proving quasiconformal extensions of holomorphic mappings f : B → C
n
to C
n
. The results are surprisingly strong. We show that the usual results from the theory, like Becker’s univalence criteria remain
true for C
1 mappings and since we use a stronger form of Loewner’s theory, we obtain results which are stronger even for holomorphic
mappings f : B → C
n
. In our main result (Theorem 4.1) we end the researches dedicated to quasiconformal extensions of K-quasiregular and holomorphic mappings f : B → C
n
to C
n
. We show that a C
1 quasiconformal map f : B → C
n
can be extended to a quasiconformal map F : C
n
→ C
n
, without any metric condition imposed to the map f. 相似文献
10.
We prove that a harmonic diffeomorphism between two Jordan domains with C
2 boundaries is a (K, K′) quasiconformal mapping for some constants K ≥ 1 and K′ ≥ 0 if and only if it is Lipschitz continuous. In this setting, if the domain is the unit disk and the mapping is normalized
by three boundary points condition we give an explicit Lipschitz constant in terms of simple geometric quantities of the Jordan
curve which surrounds the codomain and (K, K′). The results in this paper generalize and extend several recently obtained results. 相似文献
11.
Gerd Herzog 《Numerical Functional Analysis & Optimization》2013,34(5):530-538
Let E be a Banach spaces ordered by a cone K. We prove a fixed point theorem for Lipschitz continuous monotone decreasing functions f: K → K, which proves the existence of a unique fixed point in cases where the Lipschitz constant of f is bigger than 1. This fixed point theorem can be applied to Hammerstein integral equations in a quite natural way. 相似文献
12.
Ze Min Zhou 《数学学报(英文版)》2013,29(8):1543-1554
A homeomorphism w=f(z) of a domain D is called a locally quasiconformal mapping, if for each subdomain D' of D with 'D, the restriction of f(z) on D' is a quasiconformal mapping. We give some conditions for a measurable function μ(z) on the unit disc to be the complex dilatation of a locally quasiconformal mapping f which can be homeomorphically extended to the closed unit disc. 相似文献
13.
K. Strebel 《Commentarii Mathematici Helvetici》1999,74(1):143-149
A polygon P
N is the unit disk with distinguished boundary points, . An extremal quasiconformal mapping maps each polygon inscribed in onto a polygon inscribed in . Let f
N be the extremal quasiconformal mapping of P
N onto P'
N. Let K
N be its dilatation and let K
0 be the maximal dilatation of f
0. Then, evidently . The problem is, when equality holds. This is completely answered, if f
0 does not have any essential boundary points. For quadrilaterals Q and Q' = f
0
(Q) the problem is sup(M'/M) = K
0, with M and M' the moduli of Q and Q' respectively.
Received: December 23, 1997 相似文献
14.
We show that given any closed subset C of a real Banach space E, there is a continuous function f(t, x) which is Lipschitz continuous in its second variable such that the solution set of the corresponding third kind boundary value problem is homeomorphic to C (Theorem 1.1). In the special problem we give the infimum of Lipschitz constants Lf of such functions f(t, x) (Theorem 1.3). 相似文献
15.
In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while
the metric on ℂ is Euclidean. After observing that the space L{\mathcal{L}} of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on L{\mathcal{L}} that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential
norm, and determine the spectrum. We then prove that the only isometric multiplication operators on L{\mathcal{L}} are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace
of L{\mathcal{L}} we call the little Lipschitz space. 相似文献
16.
《Quaestiones Mathematicae》2013,36(5):651-663
AbstractLet G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic di?erence of the function f by the formulaQk f (x, y) := f (x + ky) + f (x ? ky) ? f (x + y) ? f (x ? y) ? 2(k2 ? 1)f (y)for all x, y ∈ G and for any integer k with k ≠ 1, ?1. In this paper, we achieve the general solution of equation Qk f (x, y) = 0, after it, we show that if Qk f is Lipschitz, then there exists a quadratic function K : G → E such that f ? K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7]. 相似文献
17.
Eva Matou
kov 《Journal of Functional Analysis》2002,190(2):507-525
Let f be a bi-Lipschitz mapping of the Euclidean ball B
n into ℓ2 with both Lipschitz constants close to one. We investigate the shape of f(B
n). We give examples of such a mapping f, which has the Lipschitz constants arbitrarily close to one and at the same time has in the supremum norm the distance at least one from every isometry of
n. 相似文献
18.
The solution set of a Dirichlet problem x″ = f(t, x), x(0) = x(1) = 0, on a Banach space E and with f satisfying a Lipschitz condition, is homeomorphic to a closed subset of E. We prove that to an closed subset C of E there is a function f with Lipschitz constant arbitrarily close to π2, such that the solution set of the corresponding Dirichlet problem is homeomorphic to C. 相似文献
19.
On an Extended Lagrange Claim 总被引:1,自引:0,他引:1
Lagrange once made a claim having the consequence that a smooth function f has a local minimum at a point if all the directional derivatives of f at that point are nonnegative. That the Lagrange claim is wrong was shown by a counterexample given by Peano. In this note, we show that an extended claim of Lagrange is right. We show that, if all the lower directional derivatives of a locally Lipschitz function f at a point are positive, then f has a strict minimum at that point. 相似文献
20.
E. Kissin D. S. Potapov V. S. Shulman F. A. Sukochev 《Functional Analysis and Its Applications》2011,45(2):157-159
It is proved that, for any Lipschitz function f(t
1, ..., t
n
) of n variables, the corresponding map f
op: (A
1, ...,A
n
) → f(A
1, ..., A
n
) on the set of all commutative n-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal S
p
, p ∈ (1,∞). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz
functions of one variable preserve domains of closed derivations with values in S
p
. It is also proved that the map f
op is Fréchet differentiable in the norm of S
p
if f is continuously differentiable. 相似文献