Singular integral operators on Riemann surfaces and elliptic differential systems
type: Item
instance of: scholarly article
title: Singular integral operators on Riemann surfaces and elliptic differential systems
zbMATH Open document ID: 0955.30033
DOI: 10.1007/BF02921715
author: Yu. L. Rodin
published in: The Journal of Geometric Analysis
publication date: 2000-06-22T00:00:00Z
review text: On some closed Riemann surface \(M\) of genus \(\geq 0\), a first-order elliptic differential system for two unknown functions \(u,v\) is considered which, by \(w=u+iv\), is rewritten in the complex form \[ \overline\partial w=q_1 \partial w+q_2 \overline{\partial w}+Aw+ B\overline w;\tag{1} \] here is assumed that \(A,B\) belong to \(L_p(M)\), \(p>2\), and \(q_1\) and \(q_2\) are measurable functions satisfying an ellipticity condition for (1). For the investigation of solutions to (1), a representation of the form \(w=\exp T_0\varphi\) with \(T_0 \varphi= -{1\over\pi} \iint{\varphi(t) d\sigma_t\over t-z}\) is utilized; hence, \[ \varphi= q_1S\varphi+ q_2{\overline {\exp T_0\varphi} \over\exp T_0 \varphi} \overline {\partial T_0\varphi} +A+B{\overline {\exp T_0\varphi}\over \exp T_0 \varphi}. \tag{2} \] First, as auxiliary tools some period relations for Abelian differentials are derived in Section 2. Then the operator \(S\) (introduced in ansatz (2)) is defined and studied in Sec. 3. Finally, in Sec. 4, the existence of a regular solution (a generalized solution without singularities) to (1) is proved. In view of (2) this involves the consideration of an integral equation and an application of a fixed-point argument.
Mathematics Subject Classification ID: 30F30
Mathematics Subject Classification ID: 30G20
Mathematics Subject Classification ID: 35J45
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 1463799
zbMATH Keywords: elliptic system of first-order
zbMATH Keywords: Riemann surface
zbMATH Keywords: Abelian differential
zbMATH Keywords: singular integral operator
Wikidata QID: Q115391203
reviewed by: M. Kracht
MaRDI profile type: Publication
cites work: Boundary value problems for pseudoanalytic functions
cites work: Q4039774
cites work: Generalized analytic functions on Riemann surfaces
cites work: Q3281267
full work available at URL: https://doi.org/10.1007/bf02921715
OpenAlex ID: W2087490538
description: scientific article; zbMATH DE number 1463799