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Báo cáo hóa học: "Research Article A Complement to the Fredholm Theory of Elliptic Systems on Bounded Domains"

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 A Complement to the Fredholm Theory of Elliptic Systems on Bounded Domains | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 637243 9 pages doi 10.1155 2009 637243 Research Article A Complement to the Fredholm Theory of Elliptic Systems on Bounded Domains Patrick J. Rabier Department of Mathematics University of Pittsburgh Pittsburgh PA 15260 USA Correspondence should be addressed to Patrick J. Rabier rabier@imap.pitt.edu Received 24 March 2009 Revised 4 June 2009 Accepted 11 June 2009 Recommended by Peter Bates We fill a gap in the p theory of elliptic systems on bounded domains by proving the pindependence of the index and null-space under minimal smoothness assumptions. This result has been known for long if additional regularity is assumed and in various other special cases possibly for a limited range of values of p. Here p-independence is proved in full generality. Copyright 2009 Patrick J. Rabier. 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. Introduction Although important issues are still being investigated today the bulk of the Fredholm theory of linear elliptic boundary value problems on bounded domains was completed during the 1960s. For pseudodifferential operators the literature is more recent and begins with the work of Boutet de Monvel 1 see also 2 for a more complete exposition. While this was the result of the work and ideas of many the most extensive treatment in the pp framework is arguably contained in the 1965 work of Geymonat 3 . This note answers a question explicitly left open in Geymonat s paper which seems to have remained unresolved. We begin with a brief partial summary of 3 in the case of a single scalar equation. Let Q be a bounded connected open subset of RN N 2 and let P denote a differential operator on Q of order 2m m 1 with complex coefficients P 2 aa x da. 1.1 a 2m Next let B B1 . Bm J be a system of boundary