An unusual selfadjoint linear partial differential operator
Authors:
W. N. Everitt, L. Markus and M. Plum
Journal:
Trans. Amer. Math. Soc. 357 (2005), 13031324
MSC (2000):
Primary 35J40, 35J67, 35P05; Secondary 32A36, 32A40, 47B25.
Published electronically:
November 4, 2004
MathSciNet review:
2115367
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of selfadjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain selfadjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region. This present paper concerns the spectral properties of one of these unusual selfadjoint operators, sometimes called the ``Harmonic'' operator. The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the selfadjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions. In the regular ordinary case the spectrum of all the selfadjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty. This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty. Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent. In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.
 1.
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)
 2.
Alberto
Alonso and Barry
Simon, The BirmanKreĭnVishik theory of selfadjoint
extensions of semibounded operators, J. Operator Theory
4 (1980), no. 2, 251–270. MR 595414
(81m:47038)
 3.
J.
M. Anderson and Loren
D. Pitt, On recurrence properties of certain lacunary series. I.
General results, J. Reine Angew. Math. 377 (1987),
65–82. MR
887400 (88i:30005a)
 4.
J.M. Anderson. Personal communication: April 2002.
 5.
Stefan
Bergman, The kernel function and conformal mapping, Second,
revised edition, American Mathematical Society, Providence, R.I., 1970.
Mathematical Surveys, No. V. MR 0507701
(58 #22502)
 6.
Nelson
Dunford and Jacob
T. Schwartz, Linear Operators. I. General Theory, With the
assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics,
Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers,
Ltd., London, 1958. MR 0117523
(22 #8302)
Nelson
Dunford and Jacob
T. Schwartz, Linear operators. Part II: Spectral theory. Self
adjoint operators in Hilbert space, With the assistance of William G.
Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons\
New YorkLondon, 1963. MR 0188745
(32 #6181)
 7.
W.
Norrie Everitt and Lawrence
Markus, Boundary value problems and symplectic algebra for ordinary
differential and quasidifferential operators, Mathematical Surveys
and Monographs, vol. 61, American Mathematical Society, Providence,
RI, 1999. MR
1647856 (2000c:34030)
 8.
W.
N. Everitt and L.
Markus, Elliptic partial differential operators and symplectic
algebra, Mem. Amer. Math. Soc. 162 (2003),
no. 770, x+111. MR 1955204
(2004d:47054), http://dx.doi.org/10.1090/memo/0770
 9.
Gaetano
Fichera, Linear elliptic differential systems and eigenvalue
problems, Lecture Notes in Mathematics, vol. 8, SpringerVerlag,
BerlinNew York, 1965. MR 0209639
(35 #536)
 10.
Avner
Friedman, Partial differential equations, Holt, Rinehart and
Winston, Inc., New YorkMontreal, Que.London, 1969. MR 0445088
(56 #3433)
 11.
G.
H. Hardy, Divergent Series, Oxford, at the Clarendon Press,
1949. MR
0030620 (11,25a)
 12.
Yitzhak
Katznelson, An introduction to harmonic analysis, John Wiley
& Sons, Inc., New YorkLondonSydney, 1968. MR 0248482
(40 #1734)
 13.
Walter
Rudin, Real and complex analysis, 3rd ed., McGrawHill Book
Co., New York, 1987. MR 924157
(88k:00002)
 14.
S. Timoshenko. Theory of plates and shells (McGrawHill, Inc., New York:1959).
 15.
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)
 16.
Kôsaku
Yosida, Functional analysis, Die Grundlehren der
Mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York;
SpringerVerlag, Berlin, 1965. MR 0180824
(31 #5054)
 17.
A.
Zygmund, Trigonometric series: Vols. I, II, Second edition,
reprinted with corrections and some additions, Cambridge University Press,
LondonNew York, 1968. MR 0236587
(38 #4882)
 1.
 R.A. Adams. Sobolov spaces (Academic Press, London and Boston, Mass.: 1978). MR 0450957 (56:9247)
 2.
 A. Alonso and B. Simon. The BirmanKreinVishik theory of selfadjoint extensions of semibounded operators. Operator Theory 4 (1980), 251170. MR 0595414 (81m:47038)
 3.
 J.M. Anderson and L.D. Pitt. On recurrence properties of certain lacunary series. I Crelle Jour. 377 (1987), 6582. MR 0887400 (88i:30005a)
 4.
 J.M. Anderson. Personal communication: April 2002.
 5.
 S. Bergman. The kernel function and conformal mapping. Mathematical Surveys and Monographs, 5 (American Mathematical Society, RI, USA: revised edition, 1970). MR 0507701 (58:22502)
 6.
 N. Dunford and J. Schwartz. Linear operators: I and II (Wiley, New York, USA: 1963). MR 0117523 (22:8302), MR 0188745 (32:6181)
 7.
 W.N. Everitt and L. Markus. Boundary value problems and symplectic algebra for ordinary and quasidifferential operators. Mathematical Surveys and Monographs, 61 (American Mathematical Society, RI, USA:1999). MR 1647856 (2000c:34030)
 8.
 W.N. Everitt and L. Markus. Elliptic partial differential operators and symplectic algebra. Memoirs of the American Mathematical Society 162 (2003), no. 770. MR 1955204 (2004d:47054)
 9.
 G. Fichera. Linear elliptic differential systems and eigenvalue problems. Lecture Notes in Mathematics, 8 (SpringerVerlag, Heidelberg: 1965). MR 0209639 (35:536)
 10.
 A. Friedman. Partial differential equations (Holt, Rinehart and Winston, New York: 1969). MR 0445088 (56:3433)
 11.
 G.H. Hardy. Divergent series (Oxford University Press, UK: 1949). MR 0030620 (11:25a)
 12.
 Y. Katznelson. An introduction to harmonic analysis (Wiley, New York, USA: 1968). MR 0248482 (40:1734)
 13.
 W. Rudin. Real and complex analysis (McGrawHill, Inc., New York:1986). MR 0924157 (88k:00002)
 14.
 S. Timoshenko. Theory of plates and shells (McGrawHill, Inc., New York:1959).
 15.
 J. Wloka. Partial differential equations (Cambridge University Press, UK: 1987). MR 0895589 (88d:35004)
 16.
 K. Yosida. Functional analysis (SpringerVerlag, Heidelberg, Germany: second printing 1966). MR 0180824 (31:5054)
 17.
 A. Zygmund. Trigonometric series. I (Cambridge Press, UK: second edition 1968). MR 0236587 (38:4882)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
35J40,
35J67,
35P05,
32A36,
32A40,
47B25.
Retrieve articles in all journals
with MSC (2000):
35J40,
35J67,
35P05,
32A36,
32A40,
47B25.
Additional Information
W. N. Everitt
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT, England, United Kingdom
Email:
w.n.everitt@bham.ac.uk
L. Markus
Affiliation:
School of Mathematics, Universty of Minnesota, Minneapolis, Minnesota 554550487
Email:
markus@math.umn.edu
M. Plum
Affiliation:
Mathematisches Institut I, Universität Karlsruhe, D76128 Karlsruhe, Germany
Email:
michael.plum@math.unikarlsruhe.de
DOI:
http://dx.doi.org/10.1090/S0002994704037195
PII:
S 00029947(04)037195
Keywords:
Linear partial differential equations,
selfadjoint partial differential operators,
spectral theory.
Received by editor(s):
April 15, 2003
Published electronically:
November 4, 2004
Dedicated:
Dedicated to Professor Johann Schröder
Article copyright:
© Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
