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

DOI:
https://doi.org/10.1090/S0002-9947-97-01859-X

MathSciNet review:
1401785

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Abstract | References | Similar Articles | Additional Information

Abstract: An index formula is proved for elliptic systems of P.D.E.'s with boundary values in a simply connected region in the plane. Let denote the elliptic operator and 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 defined on the unit cotangent bundle of 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 have invertible values. In the present paper, the index of is expressed in terms of the winding number of the determinant of .

**[AB]**M. F. Atiyah and R. Bott,*The index problem for manifolds with boundary*, Differential Analysis, Bombay Colloq., 1964, Oxford Univ. Press, London, 1964, pp. 175–186. MR**0185606****[AS]**M. F. Atiyah and I. M. Singer,*The index of elliptic operators. I*, Ann. of Math. (2)**87**(1968), 484–530. MR**0236950**, https://doi.org/10.2307/1970715**[BGR]**Joseph A. Ball, Israel Gohberg, and Leiba Rodman,*Interpolation of rational matrix functions*, Operator Theory: Advances and Applications, vol. 45, Birkhäuser Verlag, Basel, 1990. MR**1083145****[Ga]**H. Craemer,*Einige Iterations- und Relaxationsverfahren für drehsymmetrisch beanspruchte Zylinderschalen*, Österreich. Ing.-Arch.**6**(1951), 35–42 (German). MR**0045005****[GLR]**I. Gohberg, P. Lancaster, and L. Rodman,*Matrix polynomials*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Computer Science and Applied Mathematics. MR**662418****[Ro]**B. Rowley,*Matrix polynomials and the index problem for elliptic systems*, Trans. Amer. Math. Soc.**349**(1997), 3105-3148.**[Ru]**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157****[Vo]**A. I. Vol′pert,*On the index and normal solvability of boundary-value problems for elliptic systems of differential equations on the plane*, Trudy Moskov. Mat. Obšč.**10**(1961), 41–87 (Russian). MR**0144055****[We]**Juan J. Manfredi and Allen Weitsman,*On the Fatou theorem for 𝑝-harmonic functions*, Comm. Partial Differential Equations**13**(1988), no. 6, 651–668. MR**934377**, https://doi.org/10.1080/03605308808820556**[WRL]**J. T. Wloka, B. Rowley, and B. Lawruk,*Boundary value problems for elliptic systems*, Cambridge University Press, Cambridge, 1995. MR**1343490**

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Additional Information

**B. Rowley**

Affiliation:
Department of Mathematics, Champlain College, Lennoxville, Quebec, Canada

Email:
browley@abacom.com

DOI:
https://doi.org/10.1090/S0002-9947-97-01859-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