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On the Existence of a Solution in Weighted Sobolev Space to the Riemann–Hilbert Problem for an Elliptic System with Piecewise Continuous Boundary Data
Author(s) -
Mshimba Ali Seif A.
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609306018856
Subject(s) - mathematics , sobolev space , mathematical analysis , riemann–hilbert problem , holomorphic function , hölder condition , piecewise , gravitational singularity , boundary (topology) , boundary value problem , partial differential equation , hilbert space , pure mathematics
Given a first order elliptic partial differential equation we construct a solution which solves a given Riemann–Hilbert boundary value problem whose coefficients have singularities of the first kind at a finite number of some prescribed isolated points and are Holder–continuous outside those points while the free term has a finite number of integrable power singularities at some prescribed points. It is shown that the solution belongs to some weighted Sobolev space W 1, p ( D ;ρ), where the weight function ρ = ρ( z ;∂ D ) is the distance of the variable point z from the boundary ∂ D raised to a certain power. The problem is solved by first reducing it to an analogous problem for holomorphic functions. The latter is then solved. 2000 Mathematics Subject Classification 35F15, 35F30, 35J55, 35J65, 36J67.