跨越Carleson等高线的矩阵黎曼-希尔伯特问题。

IF 0.8 4区数学 Q2 MATHEMATICS

Monatshefte fur Mathematik Pub Date : 2018-01-01 Epub Date: 2017-01-28 DOI:10.1007/s00605-017-1019-0

Jonatan Lenells

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引用次数: 36

摘要

针对一类低正则性跳跃轮廓和跳跃矩阵，建立了n × n矩阵黎曼-希尔伯特问题理论。我们的基本假设是轮廓Γ是黎曼球中简单封闭Carleson曲线的有限并。特别地，允许有顶点、角和非横交点的无界轮廓。引入了L - p -Riemann-Hilbert问题的概念，建立了基本唯一性结果和Fredholm性质。我们还研究了Fredholmness对唯一可解性的意义，并证明了一个关于轮廓变形的定理。

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Matrix Riemann-Hilbert problems with jumps across Carleson contours.

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Matrix Riemann-Hilbert problems with jumps across Carleson contours.

We develop a theory of $n \times n$ -matrix Riemann-Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour $Γ$ is a finite union of simple closed Carleson curves in the Riemann sphere. In particular, unbounded contours with cusps, corners, and nontransversal intersections are allowed. We introduce a notion of $L^{p}$ -Riemann-Hilbert problem and establish basic uniqueness results and Fredholm properties. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation.

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来源期刊

Monatshefte fur Mathematik 数学-数学

CiteScore

1.60

自引率

11.10%

发文量

155

审稿时长

4-8 weeks

期刊介绍： The journal was founded in 1890 by G. v. Escherich and E. Weyr as "Monatshefte für Mathematik und Physik" and appeared with this title until 1944. Continued from 1948 on as "Monatshefte für Mathematik", its managing editors were L. Gegenbauer, F. Mertens, W. Wirtinger, H. Hahn, Ph. Furtwängler, J. Radon, K. Mayrhofer, N. Hofreiter, H. Reiter, K. Sigmund, J. Cigler. The journal is devoted to research in mathematics in its broadest sense. Over the years, it has attracted a remarkable cast of authors, ranging from G. Peano, and A. Tauber to P. Erdös and B. L. van der Waerden. The volumes of the Monatshefte contain historical achievements in analysis (L. Bieberbach, H. Hahn, E. Helly, R. Nevanlinna, J. Radon, F. Riesz, W. Wirtinger), topology (K. Menger, K. Kuratowski, L. Vietoris, K. Reidemeister), and number theory (F. Mertens, Ph. Furtwängler, E. Hlawka, E. Landau). It also published landmark contributions by physicists such as M. Planck and W. Heisenberg and by philosophers such as R. Carnap and F. Waismann. In particular, the journal played a seminal role in analyzing the foundations of mathematics (L. E. J. Brouwer, A. Tarski and K. Gödel). The journal publishes research papers of general interest in all areas of mathematics. Surveys of significant developments in the fields of pure and applied mathematics and mathematical physics may be occasionally included.