Nonlinear Riemann-Hilbert Problems and their Relationship to Extremal Problems for Holomorphic Functions

✍ Scribed by E. Wegert

• Publisher: John Wiley and Sons
• Year: 1988
• Tongue: English
• Weight: 876 KB
• Volume: 137
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19881370112

✦ Synopsis

The RIEBIANN-~~LBERT boundary-value problem (RHP) was posed by B. RIEMAXX in his dissertation in 1851. Generally speaking, it consists in the determination of a function w= SG + iv holomorphic in the complex unit disk D whose boundary values on the unit circle satisfy a given functional relation of the form (1) While the linear problem (i.e. , f ( t , u, v ) = a ( t ) ZL + b(t) v + ~( t ) ) was solved in classical papers of N. I. MTJSEHELISHVILI, I. N. VEKUA, F. D. GAKHOV, C. JACOB, and others (cf. [12], [4], [20], [lo]), one can observe great activity in the field of nonlinear RIEMAXN--BERT problems until nowadays. The spectrum of methods used in this topic is very broad and encloses, for instance, the reduction of n o d n e a r problems to linear ones, the application of iteration methods and fixed point principles, methods of monotone operator theory, imbedding methods,