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ISSN 1088-6826(e) ISSN 0002-9939(p)

Boundary differential relations for holomorphic functions on the disc

Author(s): Miran Černe; Matej Zajec
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 30E25, 35Q15
Posted: July 8, 2010
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Abstract: The existence of solutions of boundary differential relations for holomorphic functions on the disc $\Delta$ is considered. First we prove that for an arbitrary continuous positive function $\Phi$ on the complex plane $\mathbb{C}$ there exists a disc algebra function $f\in A(\Delta)$ such that $\vert f^{\prime}\vert=\Phi(f)$ on $\partial\Delta$ . Assuming some smoothness, the existence result is also proved for a quite general differential relation $\rho(\xi,f^{\prime}(\xi))=\Phi(\xi, f(\xi))$ , $\xi\in\partial\Delta$ , where $\rho$ is a defining function for a family of Jordan curves in $\mathbb{C}$ containing point 0 in its interior and $\Phi$ is a bounded positive function on $\partial\Delta\times\mathbb{C}$ .

References:

1.: H. Alexander and J. Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), 99-109. MR 779607 (86i:32025)
2.: F. G. Avkhadiev, Metrics with variable density and inverse boundary value problems (Russian), Trudy Sem. Kraev. Zadacham 25 (1990), 3-23. MR 1084789 (91k:30091)
3.: F. G. Avkhadiev and L. I. Shokleva, Generalizations of a theorem of Beurling and their applications to inverse boundary value problems, Russ. Math. (Iz. VUZ) 38 (1994), 78-81. MR 1301697 (95i:30008)
4.: F. Bauer, D. Kraus, O. Roth and E. Wegert, Beurling's free boundary value problem in conformal geometry, to appear in Israel Jour. Math.
5.: H. Begehr and M. A. Efendiev, On the asymptotics of meromorphic solutions for nonlinear Riemann-Hilbert problems, Math. Proc. Cambridge Philos. Soc. 127 (1999), 159-172. MR 1692479 (2000c:30076)
6.: A. Beurling, An extension of the Riemann mapping theorem, Acta Math. 90 (1953), 117-130. MR 0060027 (15:614e)
7.: M. Černe, Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces, Amer. J. Math. 126 (2004), 65-85. MR 2033564 (2004k:30092)
8.: M. Černe and M. Flores, Generalized Ahlfors functions, Trans. Amer. Math. Soc. 359 (2007), 671-686. MR 2255192 (2007h:30042)
9.: E. M. Čirka, Regularity of boundaries of analytic sets, Math. Sb. (NS) 117(159) (1982), 291-336; English translation: Math. USSR Sb. 45 (1983), 291-335. MR 648411 (83f:32009)
10.: E. M. Čirka, B. Coupet and A. B. Sukhov, On boundary regularity of analytic discs, Michigan Math. J. 46 (1999), 271-279. MR 1704142 (2000f:32022)
11.: K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. MR 787404 (86j:47001)
12.: P. L. Duren, Theory of spaces, Academic Press, New York and London, 1970. MR 0268655 (42:3552)
13.: M. A. Efendiev and W. L. Wendland, Nonlinear Riemann-Hilbert problems for multiply connected domains, Nonlinear Anal. 27 (1996), 37-58. MR 1390711 (97h:30057)
14.: M. A. Efendiev and W. L. Wendland, Nonlinear Riemann-Hilbert problems without transversality, Math. Nachr. 183 (1997), 73-89. MR 1434976 (98b:30038)
15.: M. A. Efendiev and W. L. Wendland, Nonlinear Riemann-Hilbert problems for doubly connected domains and closed boundary data, Topol. Methods Nonlinear Anal. 17 (2001), 111-124. MR 1846981 (2002d:30045)
16.: F. Forstnerič, Polynomial hulls of sets fibered over the circle, Indiana Univ. Math. J. 37 (1988), 869-889. MR 982834 (90g:32018)
17.: R. Fournier and S. Ruscheweyh, Free boundary value problems for analytic functions in the closed unit disk, Proc. Amer. Math. Soc. 127 (1999), 3287-3294. MR 1618666 (2000b:30059)
18.: J. Globevnik, Perturbation by analytic discs along maximal real submanifolds of $\mathbb{C}\sp N$ , Math. Z. 217 (1994), 287-316. MR 1296398 (95j:32031)
19.: G. M. Goluzin, Geometric theory of functions of a complex variable, Trans. of Math. Monographs, Vol. 26, Amer. Math. Soc., Providence, RI, 1969. MR 0247039 (40:308)
20.: C. D. Hill and G. Taiani, Families of analytic discs in $\mathbb{C}^n$ with boundaries on prescribed CR submanifold, Ann. Scuola. Norm. Sup. Pisa 5 (1978), 327-380. MR 501906 (80c:32023)
21.: M. J. Huntey, N. J. Moh and D. E. Tepper, Uniqueness theorems for some free boundary problems in univalent functions, Complex Var. Theory Appl. 48 (2003), 607-614. MR 1988687 (2004k:35421)
22.: R. Kühnau, Längentreue Randverzerrung bei analytischer Abbildung in hyperbolischer und Sphärischer Metrik, Mitt. Math. Sem. Giessen 229 (1997), 45-53. MR 1439207 (98g:30011)
23.: F. G. Maksudov and M. A. Efendiev, The nonlinear Hilbert problem for a doubly connected domain (Russian), Dokl. Akad. Nauk SSSR 290 (1986), 789-791. MR 863355 (88j:30088)
24.: Z. Slodkowski, Polynomial hulls in $\mathbb{C}^2$ and quasicircles, Ann. Scuola Norm. Sup. Pisa 16 (1989), 367-391. MR 1050332 (91m:32016)
25.: A. I. Šnirelman, The degree of a quasiruled mapping and a nonlinear Hilbert problem. Mat. Sb. 18 (1972), 373-396; English transl., Math USSR Sb. 18 (1973), 373-396. MR 0326521 (48:4865)
26.: E. Wegert, Nonlinear boundary value problems for holomorphic functions and singular integral equations, Mathematical Research 65, Akademie-Verlag, Berlin, 1992. MR 1206907 (94b:30049)
27.: E. Zeidler, Applied functional analysis (Applications to mathematical physics), Applied Mathematical Sciences 108, Springer, Berlin, 1995. MR 1347691 (96i:00005)

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