Matrix polynomials and the index problem for elliptic systems

Rowley, B.

doi:10.1090/S0002-9947-97-01860-6

Matrix polynomials and the index problem for elliptic systems
HTML articles powered by AMS MathViewer

by B. Rowley

Trans. Amer. Math. Soc. 349 (1997), 3105-3148

DOI: https://doi.org/10.1090/S0002-9947-97-01860-6

PDF | Request permission

Abstract:

The main new results of this paper concern the formulation of algebraic conditions for the Fredholm property of elliptic systems of P.D.E.’s with boundary values, which are equivalent to the Lopatinskii condition. The Lopatinskii condition is reformulated in a new algebraic form (based on matrix polynomials) which is then used to study the existence of homotopies of elliptic boundary value problems. The paper also contains an exposition of the relevant parts of the theory of matrix polynomials and the theory of elliptic systems of P.D.E.’s.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
M. S. Agranovič, Elliptic singular integro-differential operators, Uspehi Mat. Nauk 20 (1965), no. 5 (125), 3–120 (Russian). MR 0198017
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, DOI 10.1007/978-3-0348-7709-1
B. V. Fedosov, An analytic formula for the index of an elliptic boundary value problem, Mat. Sb. (N.S.) 93(135) (1974), 62–89, 151 (Russian). MR 0341537
I. M. Gel′fand, On elliptic equations, Russian Math. Surveys 15 (1960), no. 3, 113–123. MR 0123085, DOI 10.1070/RM1960v015n03ABEH004094
I. Gohberg, P. Lancaster, and L. Rodman, Matrix polynomials, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. MR 662418
Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Peter Lancaster and Miron Tismenetsky, The theory of matrices, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1985. MR 792300
Leiba Rodman, An introduction to operator polynomials, Operator Theory: Advances and Applications, vol. 38, Birkhäuser Verlag, Basel, 1989. MR 997092, DOI 10.1007/978-3-0348-9152-3
Brian Rowley, Wiener-Hopf factorization of operator polynomials, Integral Equations Operator Theory 3 (1980), no. 3, 437–462. MR 580717, DOI 10.1007/BF01701501
—, An index formula for elliptic systems in the plane, Trans. Amer. Math. Soc. 349 (1997), 3149–3179.
R. T. Seeley, Singular integrals and boundary value problems, Amer. J. Math. 88 (1966), 781–809. MR 209915, DOI 10.2307/2373078
J. Wloka, Partial differential equations, Cambridge University Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M. J. Thomas. MR 895589, DOI 10.1017/CBO9781139171755
J. T. Wloka, B. Rowley, and B. Lawruk, Boundary value problems for elliptic systems, Cambridge University Press, Cambridge, 1995. MR 1343490, DOI 10.1017/CBO9780511662850