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Another Approach to Liouville Theorem

✍ Scribed by B. Bojarski; T. Iwaniec

Publisher
John Wiley and Sons
Year
1982
Tongue
English
Weight
417 KB
Volume
107
Category
Article
ISSN
0025-584X

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✦ Synopsis

Below we give a direct proof of LIOG'VTLLE theorem on conformal mapping3 in domains of R". n z 3. We accept the following analytical definition of conformal-Ity which is known in the theory of quasiregular mappings. Let R be an open suhset of R".

Definition. The mapping f :

it satisfies the following conditions

(1) ulrnost etur!ywhere in Q. a) f belongs to the SOBOLEV space M'A,lm(12) 1)) o*fof= J*/"E, J = J ( Z , f ) g o Here Df is the JACOBI matrix of the map f , Of= 7 , D*f is the matrix transposed to Df and J = J ( z . f ) denotes the JACoBran o f f . (3 Theorein (LIOL-VILLE). Let f2 be rr connected domain in Rn. Every genercrl cnnformnl mapping for n 2 3 is either ronstant or if is ci restriction to l2 of c( MMOBIC-S fro nuformdion.

📜 SIMILAR VOLUMES

A Lot of “Counterexamples” to Liouville'
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✍ Luis Bernal-González 📂 Article 📅 1996 🏛 Elsevier Science 🌐 English ⚖ 122 KB

We prove in this paper that, given ␣ g 0, 1r2 , there exists a linear manifold M of entire functions satisfying that M is dense in the space of all entire functions Ž< < ␣ . Ž j. Ž . and, in addition, lim exp z f z s0 on any plane strip for every f g M z ª ϱ and for every derivation index j. Moreove