Existence of solution in weighted Sobolev spaceWn,p(D; dμ(z; ∂D)) to Riemann–Hilbert problem with piece-wise continuous boundary data for non-homogeneous polyanalytic equation of ordern

2005 ◽  
Vol 50 (7-11) ◽  
pp. 631-644
Author(s):  
Ali Seif Mshimba *
Keyword(s):  
Boundary Data ◽  
Hilbert Problem ◽  
Existence Of Solution ◽  
Continuous Boundary ◽  
Riemann Hilbert Problem

scholarly journals On the Riemann-Hilbert problem in multiply connected domains

2016 ◽  
Vol 14 (1) ◽  
pp. 13-18 ◽  
Author(s):  
Vladimir Ryazanov
Keyword(s):  
Boundary Data ◽  
Hilbert Problem ◽  
Multiply Connected Domains ◽  
Natural Parameter ◽  
Multiply Connected ◽  
Jordan Curves ◽  
Measurable Coefficients ◽  
Asymptotic Values ◽  
Riemann Hilbert Problem ◽  
Number Of Branches

AbstractWe proved the existence of multivalent solutions with the infinite number of branches for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthermore, it is shown that the dimension of the spaces of these solutions is infinite.

scholarly journals The hyperbolic Ernst equation in a triangular domain

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Jonatan Lenells ◽  
Julian Mauersberger
Keyword(s):  
Gravitational Waves ◽  
Boundary Data ◽  
Hilbert Problem ◽  
Goursat Problem ◽  
Theory Of Relativity ◽  
Singular Behavior ◽  
Integrable Structure ◽  
Ernst Equation ◽  
Triangular Domain ◽  
Riemann Hilbert Problem

AbstractThe collision of two plane gravitational waves in Einstein’s theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann–Hilbert problem. The formulation of the Riemann–Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform. Our results are valid also for boundary data whose derivatives are unbounded at the triangle’s corners—this level of generality is crucial for the application to colliding gravitational waves. Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann–Hilbert formalism can be explicitly inverted at the boundary. In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary.

scholarly journals On the Modular Operator of Mutli-component Regions in Chiral CFT

2021 ◽  
Author(s):  
Stefan Hollands
Keyword(s):  
Field Theory ◽  
Integral Equation ◽  
Hilbert Problem ◽  
Matrix Elements ◽  
New Approach ◽  
Conformal Field ◽  
The Matrix ◽  
Kms Condition ◽  
Modular Operator ◽  
Riemann Hilbert Problem

AbstractWe introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

scholarly journals Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa II: Riemann–Hilbert Problem

2015 ◽  
Vol 336 (1) ◽  
pp. 337-380 ◽  
Author(s):  
Martin A. Guest ◽  
Alexander R. Its ◽  
Chang-Shou Lin
Keyword(s):  
Hilbert Problem ◽  
Riemann Hilbert Problem

scholarly journals Integral Harnack inequality

1985 ◽  
Vol 26 (2) ◽  
pp. 115-120 ◽  
Author(s):  
Murali Rao
Keyword(s):  
Euclidean Space ◽  
Hausdorff Measure ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Convex Domains ◽  
Dimensional Hausdorff Measure ◽  
Image Position ◽  
Continuous Boundary ◽  
Integral Version

Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f thenwhere ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc.

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