TAILIEUCHUNG - Báo cáo hóa học: " Research Article Removable Singularities of and Quasiregular Mappings"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Removable Singularities of and Quasiregular Mappings | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID 61602 9 pages doi 2007 61602 Research Article Removable Singularities of VJ -Differential Forms and Quasiregular Mappings Olli Martio Vladimir Miklyukov and Matti Vuorinen Received 14 May 2006 Revised 6 September 2006 Accepted 20 September 2006 Recommended by Ugo Pietro Gianazza A theorem on removable singularities of WJ-differential forms is proved and applied to quasiregular mappings. Copyright 2007 Olli Martio et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Main theorem We recall some facts on differential forms and quasiregular mappings. Our notation is as in 1 . Let M be a Riemannian manifold of the class C3 dimM n without boundary. Each differential form a can be written in terms of the local coordinates x1 . xn as the linear combination a ai1 .ịkdxị1 A A dxịk. 1 i1 . ik n Let a be a differential form defined on an open set D c M. If D is a class of functions defined on D then we say that the differential form a is in this class provided that G D . For instance the differential form a is in the class Lp D if all its coefficients are in this class. A differential form a of degree k on the manifold M with coefficients ai1 .ik G Lpoc M is called weakly closed if for each differential form p degp k 1 with compact support suppp m G M p 0 in M and with coefficients in the class wq loc 1 p 1 q 1 1 p q TO we have f a ôp M 0. M 2 Boundary Value Problems Here the operator and the exterior differentiation d define the codifferential operator 8 by the formula 8a -1 k -1d a for a differential form a of degree k. Clearly 8a is a differential form of degree k - 1. For smooth differential forms a condition agrees with the traditional condition of closedness da 0. For an arbitrary simple form of .

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