Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem
We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation (PDE) Δ u=4 e2u and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary beha...
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| Published in: | Journal of the London Mathematical Society Vol. 77; no. 1; pp. 183 - 202 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Oxford University Press
01-02-2008
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| Online Access: | Get full text |
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| Summary: | We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation (PDE) Δ u=4 e2u and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary behaviour. For instance, we show that for every Blaschke sequence {zj} in the unit disk there is always a Blaschke product with {zj} as its set of critical points. Our work is closely related to the Berger--Nirenberg problem in differential geometry. |
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| Bibliography: | istex:109A42CBC603DFEBAB5CE7C5320FE4CAF00F06C3 ark:/67375/HXZ-4SWGXDR6-4 ArticleID:jdm095 2000 Mathematics Subject Classification 30D50, 35J65 (primary), 53A30, 30F45 (secondary). 2000 The first author was supported by an HWP scholarship. The second author received partial support from the German‐‐Israeli Foundation (grant G‐809‐234.6/2003). Mathematics Subject Classification 30D50, 35J65 (primary), 53A30, 30F45 (secondary). |
| ISSN: | 0024-6107 1469-7750 |
| DOI: | 10.1112/jlms/jdm095 |