An index formula for elliptic systems
in the plane
Trans. Amer. Math. Soc. 349 (1997), 3149-3179
Primary 35J40, 35J55, 15A22
Full-text PDF Free Access
Similar Articles |
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 .
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
F. Atiyah and I.
M. Singer, The index of elliptic operators. I, Ann. of Math.
(2) 87 (1968), 484–530. MR 0236950
A. Ball, Israel
Gohberg, and Leiba
Rodman, Interpolation of rational matrix functions, Operator
Theory: Advances and Applications, vol. 45, Birkhäuser Verlag,
Basel, 1990. MR
Craemer, Einige Iterations- und Relaxationsverfahren für
drehsymmetrisch beanspruchte Zylinderschalen, Österreich.
Ing.-Arch. 6 (1951), 35–42 (German). MR 0045005
Lancaster, and L.
Rodman, Matrix polynomials, Academic Press Inc. [Harcourt
Brace Jovanovich Publishers], New York, 1982. Computer Science and Applied
B. Rowley, Matrix polynomials and the index problem for elliptic systems, Trans. Amer. Math. Soc. 349 (1997), 3105-3148.
Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book
Co., New York, 1987. MR 924157
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
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
T. Wloka, B.
Rowley, and B.
Lawruk, Boundary value problems for elliptic systems,
Cambridge University Press, Cambridge, 1995. MR 1343490
- M.F. Atiyah, R. Bott, The index problem for manifolds with boundary, Bombay Colloquium on Differential Analysis, Oxford Univ. Press, 1964, pp. 175-186. MR 32:3069
- M.F. Atiyah, I.M. Singer, The index of elliptic operators I, Ann. of Math. 87 (1968), 484-530. MR 38:5243
- J.A. Ball, I. Gohberg, L. Rodman, Interpolation of Rational Matrix Functions, Operator Theory: Advances and Applications 45, Birkhäuser Verlag, Basel and Boston, 1990. MR 92m:47027
- F.D. Gakhov, Boundary Value Problems, Dover Publications, New York, 1990. MR 45005
- I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. MR 84c:15012
- B. Rowley, Matrix polynomials and the index problem for elliptic systems, Trans. Amer. Math. Soc. 349 (1997), 3105-3148.
- W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill, New York, 1987. MR 88k:00002
- A.I. Volpert, On the index and the normal solvability of boundary value problems for elliptic systems of differential equations in the plane, Trudy Mos. Mat. Obs. 10 (1961), 41-87 (Russian). MR 26:1603
- W.L. Wendland, Elliptic Systems in the Plane, Pitman, London, 1979. MR 89h:35053
- J. Wloka, B. Rowley, B. Lawruk, Boundary Value Problems for Elliptic Systems, Cambridge University Press, 1995. MR 96f:35003
Retrieve articles in Transactions of the American Mathematical Society
with MSC (1991):
Retrieve articles in all journals
with MSC (1991):
Department of Mathematics, Champlain College, Lennoxville, Quebec, Canada
Elliptic boundary value problems,
Received by editor(s):
August 16, 1994
© Copyright 1997 American Mathematical Society