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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An index formula for elliptic systems
in the plane


Author: B. Rowley
Journal: Trans. Amer. Math. Soc. 349 (1997), 3149-3179
MSC (1991): Primary 35J40, 35J55, 15A22
MathSciNet review: 1401785
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Abstract: An index formula is proved for elliptic systems of P.D.E.'s with boundary values in a simply connected region $\Omega $ in the plane. Let $\mathcal {A}$ denote the elliptic operator and $\mathcal {B}$ the boundary operator. In an earlier paper by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the boundary, a square matrix function $\Delta ^{+}_{{\mathcal {B}}}$ defined on the unit cotangent bundle of $\partial \Omega $ was constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the condition that the function $\Delta ^{+}_{{\mathcal {B}}}$ have invertible values. In the present paper, the index of $({\mathcal {A}},{\mathcal {B}})$ is expressed in terms of the winding number of the determinant of $\Delta ^{+}_{{\mathcal {B}}}$.


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Additional Information

B. Rowley
Affiliation: Department of Mathematics, Champlain College, Lennoxville, Quebec, Canada
Email: browley@abacom.com

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01859-X
PII: S 0002-9947(97)01859-X
Keywords: Elliptic boundary value problems, matrix polynomials, index formula, Riemann-Hilbert problem
Received by editor(s): August 16, 1994
Article copyright: © Copyright 1997 American Mathematical Society