共查询到20条相似文献,搜索用时 46 毫秒
1.
Dr. Dieter Gaier 《Results in Mathematics》1986,10(1-2):66-81
We consider an area problem for quadrilaterals Q. The geometry of Q is partly prescribed and partly free; the module is to be fixed, and its area is to be minimized. If Q is sufficiently simple, we can solve the problem explicitly by using a modified Schwarz-Christoffel integral. The free boundary arc is represented by an integral. The corresponding area problem for doubly connected domains can be reduced to the area problem for quadrilaterals, if the given boundary component is a regular polygon. We give results of numerical experiments. 相似文献
2.
We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given. 相似文献
3.
R. Laugesen 《Constructive Approximation》1994,10(4):523-554
In this paper we investigate theoretically an approximation technique for avoiding the crowding phenomenon in numerical conformal mapping. The method applies to conformal maps from rectangles to “long quadrilaterals,” i.e., Jordan domains bounded by two parallel straight lines and two Jordan arcs, where the two arcs are far apart. We require that these maps take the four corners of the rectangle to the four corners of the quadrilateral. Our main theorem tackles a conformal mapping problem for doubly connected domains, and we derive from this our results for quadrilaterals. As a corollary, we extend the “domain decomposition” mapping technique of Papamichael and Stylianopoulos. Similar results hold for the inverse maps, from quadrilaterals to rectangles. 相似文献
4.
Matthew Badger 《Mathematische Zeitschrift》2012,270(1-2):241-262
We show the David–Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As one consequence every Wolff snowflake has infinite surface measure. 相似文献
5.
The Neumann problem as formulated in Lipschitz domains with Lp boundary data is solved for harmonic functions in any compact polyhedral domain of ℝ4 that has a connected 3-manifold boundary. Energy estimates on the boundary are derived from new polyhedral Rellich formulas
together with a Whitney type decomposition of the polyhedron into similar Lipschitz domains. The classical layer potentials
are thereby shown to be semi-Fredholm. To settle the onto question a method of continuity is devised that uses the classical 3-manifold theory of E. E. Moise in order to untwist the
polyhedral boundary into a Lipschitz boundary. It is shown that this untwisting can be extended to include the interior of
the domain in local neighborhoods of the boundary. In this way the flattening arguments of B. E. J. Dahlberg and C. E. Kenig
for the H1at Neumann problem can be extended to polyhedral domains in ℝ4. A compact polyhedral domain in ℝ6 of M. L. Curtis and E. C. Zeeman, based on a construction of M. H. A. Newman, shows that the untwisting and flattening techniques
used here are unavailable in general for higher dimensional boundary value problems in polyhedra. 相似文献
6.
Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric
trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝn. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality
and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved
for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested
and left as an open problem in [4]. Many variants will be investigated in a companion article [10]. 相似文献
7.
This paper is concerned with the eigenvalues of the Neumann Laplacian on various classes of domains of given measure: simply‐connected Lipschitz planar domains, n‐sided planar polygons and smooth N‐dimensional domains. In each case, we consider some quantities involving low eigenvalues of the Neumann Laplacian for which we obtain new inequalities. Moreover, we sharpen a universal bound derived by M. Ashbaugh and R. Benguria for sum of reciprocal of Neumann eigenvalues. Our investigations make use of some properties of conformal mappings, Bessel functions, symmetric domains or some isoperimetric inequalities for moments of inertia. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
8.
We establish Lp regularity for the Szegö and Bergman projections associated to a simply connected planar domain in any of the following classes: vanishing chord arc; Lipschitz; Ahlfors-regular; or local graph (for the Szegö projection to be well defined, the local graph curve must be rectifiable). As applications, we obtain Lp regularity for the Riesz transforms, as well as Sobolev space regularity for the non-homogeneous Dirichlet problem associated to any of the domains above and, more generally, to an arbitrary proper simply connected domain in the plane. 相似文献
9.
Mirela Kohr Massimo Lanza de Cristoforis Sergey E. Mikhailov Wolfgang L. Wendland 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2016,67(5):116
The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the nonlinear Darcy–Forchheimer–Brinkman system and the linear Stokes system in two complementary Lipschitz domains in \({\mathbb{R}^{3}}\), one of them is a bounded Lipschitz domain \({\Omega}\) with connected boundary, and the other one is the exterior Lipschitz domain \({\mathbb{R}^{3} \setminus \overline{\Omega }}\). We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces. 相似文献
10.
We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that “consistent,” conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ=6. 相似文献
11.
We consider the conformal mapping of ``strip-like' domains and derive a number of asymptotic results for computing the conformal
modules of an associated class of quadrilaterals. These results are then used for the following two purposes: (a) to estimate
the error of certain engineering formulas for measuring resistance values of integrated circuit networks; and (b) to compute
the modules of complicated quadrilaterals of the type that occur frequently in engineering applications.
April 17, 1997. Date revised: September 10, 1997. 相似文献
12.
ABSTRACT We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stronger results for a special class of “almost convex” domains, which apply to domains with corners. 相似文献
13.
E. M. Dyn'kin 《Journal of Mathematical Sciences》1986,34(6):2060-2073
The classical estimates for analytic functions in a disc are carried over to functions which are analytic in Jordan domains whose boundaries are Lipschitz and Radon curves. Singular integrals, maximal estimates, and Lusin's inequality are considered. Analytic functions, the moduli of whose boundary values satisfy the conditions of B. Muckenhoupt are studied in detail. Properties of conformal maps and their level lines connected with the Muckenhoupt conditions on the moduli of the boundary derivatives are considered. This consideration lets one give real proofs of various distortion theorems for conformal maps of Lipschitz domains.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 70–90, 1977. 相似文献
14.
A Carnot group is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study the notions of intrinsic graphs and of intrinsic Lipschitz graphs within Carnot groups. Intrinsic Lipschitz graphs are the natural local analogue inside Carnot groups of Lipschitz submanifolds in Euclidean spaces, where “natural” emphasizes that the notion depends only on the structure of the algebra. Intrinsic Lipschitz graphs unify different alternative approaches through Lipschitz parameterizations or level sets. We provide both geometric and analytic characterizations and a clarifying relation between these graphs and Rumin’s complex of differential forms. 相似文献
15.
Two-dimensional affine A-nets in $3$ -space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The defining property of A-nets is planarity of vertex stars, hence elementary quadrilaterals of a generic A-net are skew. The present article deals with the extension of A-nets to differentiable surfaces, by gluing hyperboloid surface patches into the skew quadrilaterals. The obtained surfaces, named “hyperbolic nets”, are a novel, piecewise smooth discretization of surfaces parametrized along asymptotic lines. A simply connected affine A-net can be extended to a hyperbolic net if all quadrilateral strips are “equi-twisted”. The geometric condition of equi-twist implies the combinatorial property, that all inner vertices of the A-net have to be of even degree. If an A-net can be extended to a hyperbolic net, then there exists a 1-parameter family of such extensions. It is briefly explained how the generation of hyperbolic nets can be implemented on a computer. We use a projective model of Plücker line geometry in order to describe A-nets and hyperboloids. 相似文献
16.
Vladimir Kozlov 《Journal of Differential Equations》2006,230(2):532-555
We consider the first eigenvalue of the Dirichlet-Laplacian in three cases: C1,1-domains, Lipschitz domains, and bounded domains without any smoothness assumptions. Asymptotic formula for this eigenvalue is derived when domain subject arbitrary perturbations. For Lipschitz and arbitrary nonsmooth domains, the leading term in the asymptotic representation distinguishes from that in the Hardamard formula valid for smooth perturbations of smooth domains. For asymptotic analysis we propose and prove an abstract theorem demonstrating how eigenvalues vary under perturbations of both operator in Hilbert space and Hilbert space itself. This abstract theorem is of independent interest and has substantially broader field of applications. 相似文献
17.
Daniel Daners 《Archiv der Mathematik》2009,92(1):57-69
It is proved that elliptic boundary value problems in divergence form can be written in many equivalent forms. This is used
to prove regularity properties and maximum principles for problems with Robin boundary conditions with negative or indefinite
boundary coefficient on Lipschitz domains by rewriting them as a problem with positive coefficient. It is also shown that
such methods cannot be applied to domains with an outward pointing cusp. Applications to the regularity of the harmonic Steklov
eigenfunctions on Lipschitz domains are given.
Received: 26 June 2008; Revised: 12 September 2008 相似文献
18.
In this paper we prove new results for p harmonic functions, p≠2, 1<p<∞, in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of the boundary of a Lipschitz domain, with constants only depending on p,n and the Lipschitz constant of the domain. For p capacitary functions, in starlike Lipschitz ring domains, we prove an even stronger result, Theorem 2, showing that the ratio is Hölder continuous up to the boundary. Moreover, for p capacitary functions in starlike Lipschitz ring domains we prove, Theorems 3 and 4, appropriate extensions to p≠2, 1<p<∞, of famous results of Dahlberg [12] and Jerison and Kenig [25] on the Poisson kernel associated to the Laplace operator (i.e. p=2). 相似文献
19.
An important open problem in geometric complex analysis is to establish some algorithms for explicit determination of the basic functionals intrinsically connected with conformal and quasiconformal mappings such as their Teichmüller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficient. This problem has not been solved even for generic quadrilaterals. We provide a restricted solution of the problem for unbounded rectilinear polygons.
相似文献20.
We study Hardy spaces of solutions to the conjugate Beltrami equation with Lipschitz coefficient on Dini-smooth simply connected planar domains, in the range of exponents 1<p<∞. We analyse their boundary behaviour and certain density properties of their traces. We derive on the way an analog of the Fatou theorem for the Dirichlet and Neumann problems associated with the equation div(σ∇u)=0 with Lp boundary data. 相似文献