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On Boundary-Layer-Type Solutions of Nonlinear Riemann-Hilbert Problems and Related Geometric Structures

M. A. Efendiev and W. L. Wendland
Proceedings: Mathematical, Physical and Engineering Sciences
Vol. 458, No. 2019 (Mar. 8, 2002), pp. 539-562
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/3067432
Page Count: 24
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On Boundary-Layer-Type Solutions of Nonlinear Riemann-Hilbert Problems and Related Geometric Structures
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Abstract

We consider a class of holomorphic solutions to nonlinear Riemann-Hilbert problems (RHPp) for arbitrary, p-connected domains. The boundary condition bundle here violates the so-called 'zero' conditions for the boundary curves. We present sufficient conditions for these boundary curves which provide existence of a countable number of holomorphic solutions depending on their number of zeros and having additional surprising properties. In particular, we prove the existence of holomorphic solutions of the nonlinear (RHPp), whose 'distribution of zeros' is defined by an arbitrarily given absolute positive measure. We emphasize that, in contrast to many global existence results based on topological methods, here we present constructive proofs which might also be relevant for the numerical computation of solutions. Moreover, we introduce the 'quasi-cylindrical structure' in a Banach space related to the violation of the so-called 'zero' conditions. It should be noted that our quasi-cylindrical structure appears rather naturally whenever analytically given nonlinear pseudodifferential operators are considered on spaces of sufficiently smooth functions.

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