On the Riemann–Hilbert Problem in the Domain with a Nonsmooth Boundary

Abstract The following Riemann–Hilbert problem is solved: find an analytical function Φ from the Smirnov class Ep (D), whose angular boundary values satisfy the condition Re[(a(t) + ib(t))Φ+ (t)] = ƒ(t). The boundary Γ of the domain D is assumed to be a piecewise smooth curve whose nonintersecting Lyapunov arcs form, with respect to D, the inner angles with values νkπ, 0 < νk ≤ 2.

Download Full-text

The Riemann–Hilbert Problem in a Domain with Piecewise Smooth Boundaries in Weight Classes of Cauchy Type Integrals with a Density from Variable Exponent Lebesgue Spaces

Georgian Mathematical Journal ◽

10.1515/gmj.2009.737 ◽

2009 ◽

Vol 16 (4) ◽

pp. 737-755 ◽

Cited By ~ 1

Author(s):

Vakhtang Kokilashvili ◽

Vakhtang Paatashvili

Keyword(s):

Variable Exponent ◽

Hilbert Problem ◽

Domain Boundary ◽

Smooth Curve ◽

Piecewise Smooth ◽

Cauchy Type ◽

Lebesgue Spaces ◽

Variable Exponent Lebesgue Spaces ◽

Cauchy Type Integrals ◽

Riemann Hilbert Problem

Abstract The Riemann–Hilbert problem for an analytic function is solved in weighted classes of Cauchy type integrals in a simply connected domain not containing 𝑧 = ∞ and having a density from variable exponent Lebesgue spaces. It is assumed that the domain boundary is a piecewise smooth curve. The solvability conditions are established and solutions are constructed. The solution is found to essentially depend on the coefficients from the boundary condition, the weight, space exponent values at the angular points of the boundary curve and also on the angle values. The non-Fredholmian case is investigated. An application of the obtained results to the Neumann problem is given.

Download Full-text

The elliptic sinh-Gordon equation in a semi-strip

Advances in Nonlinear Analysis ◽

10.1515/anona-2016-0206 ◽

2017 ◽

Vol 8 (1) ◽

pp. 533-544 ◽

Cited By ~ 2

Author(s):

Guenbo Hwang

Keyword(s):

Gordon Equation ◽

Hilbert Problem ◽

Lax Pair ◽

Inverse Scattering Transform ◽

Boundary Values ◽

Spectral Functions ◽

Scattering Transform ◽

The Matrix ◽

Fokas Method ◽

Riemann Hilbert Problem

Abstract We study the elliptic sinh-Gordon equation posed in a semi-strip by applying the so-called Fokas method, a generalization of the inverse scattering transform for boundary value problems. Based on the spectral analysis for the Lax pair formulation, we show that the spectral functions can be characterized from the boundary values. We express the solution of the equation in terms of the unique solution of the matrix Riemann–Hilbert problem whose jump matrices are defined by the spectral functions. Moreover, we derive the global algebraic relation that involves the boundary values. In addition, it can be verified that the solution of the elliptic sinh-Gordon equation posed in the semi-strip exists if the spectral functions defined by the boundary values satisfy this global relation.

Download Full-text

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

Communications in Mathematical Physics ◽

10.1007/s00220-021-04054-6 ◽

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.

Download Full-text