Matrix polynomials and the index problem for elliptic systems
Author:
B. Rowley
Journal:
Trans. Amer. Math. Soc. 349 (1997), 31053148
MSC (1991):
Primary 35J45, 35J55, 15A22
MathSciNet review:
1401786
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Abstract 
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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.
 [Ad]
Robert
A. Adams, Sobolev spaces, Academic Press [A subsidiary of
Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1975. Pure and
Applied Mathematics, Vol. 65. MR 0450957
(56 #9247)
 [ADN]
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
0125307 (23 #A2610)
S.
Agmon, A.
Douglis, and L.
Nirenberg, Estimates near the boundary for solutions of elliptic
partial differential equations satisfying general boundary conditions.
II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 0162050
(28 #5252)
 [Ag]
M.
S. Agranovič, Elliptic singular integrodifferential
operators, Uspehi Mat. Nauk 20 (1965), no. 5
(125), 3–120 (Russian). MR 0198017
(33 #6176)
 [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 (92m:47027)
 [Fe]
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
(49 #6285)
 [Ge]
I.
M. Gel′fand, On elliptic equations, Russian Math.
Surveys 15 (1960), no. 3, 113–123. MR 0123085
(23 #A416)
 [GLR]
I.
Gohberg, P.
Lancaster, and L.
Rodman, Matrix polynomials, Academic Press, Inc. [Harcourt
Brace Jovanovich, Publishers], New YorkLondon, 1982. Computer Science and
Applied Mathematics. MR 662418
(84c:15012)
 [Ho]
Lars
Hörmander, The analysis of linear partial differential
operators. III, Grundlehren der Mathematischen Wissenschaften
[Fundamental Principles of Mathematical Sciences], vol. 274,
SpringerVerlag, Berlin, 1985. Pseudodifferential operators. MR 781536
(87d:35002a)
 [Lo]
Ya.
B. Lopatinskiĭ, On a method of reducing boundary problems
for a system of differential equations of elliptic type to regular integral
equations, Ukrain. Mat. Ž. 5 (1953),
123–151 (Russian). MR 0073828
(17,494b)
 [LT]
Peter
Lancaster and Miron
Tismenetsky, The theory of matrices, 2nd ed., Computer Science
and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1985. MR 792300
(87a:15001)
 [R]
Leiba
Rodman, An introduction to operator polynomials, Operator
Theory: Advances and Applications, vol. 38, Birkhäuser Verlag,
Basel, 1989. MR
997092 (90k:47032)
 [Ro1]
Brian
Rowley, WienerHopf factorization of operator polynomials,
Integral Equations Operator Theory 3 (1980), no. 3,
437–462. MR
580717 (81i:47023), http://dx.doi.org/10.1007/BF01701501
 [Ro2]
, An index formula for elliptic systems in the plane, Trans. Amer. Math. Soc. 349 (1997), 31493179.
 [Se]
R.
T. Seeley, Singular integrals and boundary value problems,
Amer. J. Math. 88 (1966), 781–809. MR 0209915
(35 #810)
 [Wl]
J.
Wloka, Partial differential equations, Cambridge University
Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M.
J. Thomas. MR
895589 (88d:35004)
 [WRL]
J.
T. Wloka, B.
Rowley, and B.
Lawruk, Boundary value problems for elliptic systems,
Cambridge University Press, Cambridge, 1995. MR 1343490
(96f:35003)
 [Ad]
 R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 56:9247
 [ADN]
 S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 12 (1959), 623727; II, Comm. Pure Appl. Math. 17 (1964), 3592. MR 23:A2610; MR 28:5252
 [Ag]
 M.S. Agranovi\v{c}, Elliptic singular integrodifferential operators, Russian Math. Surveys 20 (5) (1965), 1122. MR 33:6176
 [BGR]
 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
 [Fe]
 B.V. Fedosov, An analytic formula for the index of an elliptic boundaryvalue problem, Math. USSR Sbornik 22 (1974), 6190. MR 49:6285
 [Ge]
 I. M. Gel'fand, On elliptic equations, Russian Math. Surveys 15 (3) (1960), 113123. MR 23:A416
 [GLR]
 I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. MR 84c:15012
 [Ho]
 L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. III, SpringerVerlag, New York and Berlin, 1985. MR 87d:35002a
 [Lo]
 Ya. B. Lopatinskii, On a method of reducing boundary value problems for a system of differential equations of elliptic type to regular integral equations, Amer. Math. Soc. Transl. (Ser. 2) 89 (1970), 149183. MR 17:494b
 [LT]
 P. Lancaster, M. Tismenetsky, The Theory of Matrices with Applications, Academic Press, New York, 1985. MR 87a:15001
 [R]
 L. Rodman, An Introduction to Operator Polynomials, OT 38, Birkhäuser Verlag, Basel and Boston, 1989. MR 90k:47032
 [Ro1]
 B. Rowley, WienerHopf factorization of operator polynomials, Integral Equations and Operator Theory 3 (3) (1980), 437462. MR 81i:47023
 [Ro2]
 , An index formula for elliptic systems in the plane, Trans. Amer. Math. Soc. 349 (1997), 31493179.
 [Se]
 R. Seeley, Singular integrals and boundary value problems, Amer. J. Math. 88 (1966), 781809. MR 35:810
 [Wl]
 J. Wloka, Partial Differential Equations, Cambridge University Press, 1987. MR 88d:35004
 [WRL]
 J. Wloka, B. Rowley, B. Lawruk, Boundary Value Problems for Elliptic Systems, Cambridge University Press, 1995. MR 96f:35003
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Additional Information
B. Rowley
Affiliation:
Department of Mathematics, Champlain College, Lennoxville, Quebec, Canada
Email:
browley@lennox.champlaincollege.qc.ca
DOI:
http://dx.doi.org/10.1090/S0002994797018606
PII:
S 00029947(97)018606
Keywords:
Elliptic boundary value problems,
matrix polynomials,
Lopatinskii condition,
Fredholm property
Received by editor(s):
August 16, 1994
Received by editor(s) in revised form:
February 12, 1996
Additional Notes:
The author wishes to acknowledge that the abstract, the introduction and parts of §§2 and 3 were revised due to the helpful remarks and suggestions of the referee.
Article copyright:
© Copyright 1997
American Mathematical Society
