Second order elliptic equations with degenerate weight
Author:
W. Allegretto
Journal:
Proc. Amer. Math. Soc. 107 (1989), 989998
MSC:
Primary 35J10; Secondary 35P15, 47F05
MathSciNet review:
977929
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Abstract 
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Abstract: We consider the eigenvalue problem: , in a smooth bounded domain . We allow to have negative spectrum and assume in in a subdomain of . Under suitable regularity conditions, we establish several results for the spectrum of this problem. In particular, we give: a min.max. formula for ; a precise estimate on the number of negative ; an estimate for the location of negative . An example concludes the paper.
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 W. Allegretto and A. Mingarelli, On the nonexistence of positive solutions for a Schrödinger equation with an indefinite weight function, C.R. Math. Rep. Acad. Sci. Canada 8 (1986), 6973. MR 827120 (87j:35154)
 [2]
 , Boundary problems of the second order with an indefinite weight function, preprint.
 [3]
 P. Binding and P. Browne, Spectral properties of two parameter eigenvalue problems II, Proc. Royal Soc. Edinb. 106A (1987), 3951. MR 899939 (88m:47036)
 [4]
 R. Courant and D. Hilbert, Methods of mathematical physics, Vol. I, Interscience, New York, 1953. MR 0065391 (16:426a)
 [5]
 W. N. Everitt, M. Kwong and A. Zettl, Oscillation of eigenfunctions of weighted regular SturmLiouville problems, J. London Math. Soc. 27 (1983), 106120. MR 686509 (84g:34035)
 [6]
 , Differential operators and quadratic inequalities with a degenerate weight, J. Math. Anal. Appl. 98 (1984), 378399. MR 730514 (85m:34036)
 [7]
 D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd Edition, Springer, Berlin/New York, 1983. MR 737190 (86c:35035)
 [8]
 I.C. Gohberg and M. G. Krein, Theory and applications of Volterra operators in Hilbert space, Trans, of Math. Mono., Vol. 24, American Mathematical Society, 1970. MR 0264447 (41:9041)
 [9]
 E. L. Ince, Ordinary differential equations, Dover, New York, 1956. MR 0010757 (6:65f)
 [10]
 M. Reed and B. Simon, Methods of modern mathematical physics, Vol. IV, Academic Press, New York, 1978. MR 0493421 (58:12429c)
 [11]
 F. Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach, New York, 1969. MR 0240668 (39:2014)
 [12]
 F. Reisz and B. Sz.Nagy, Functional analysis, Ungar Publishing, New York, 1955. MR 0071727 (17:175i)
 [13]
 S. L. Sobolev, Applications of functional analysis in mathematical physics, Trans, of Math. Mono., Vol. 7, American Mathematical Society, 1963. MR 0165337 (29:2624)
 [14]
 R. Vyborny, Continuous dependence of eigenvalues on the domain, Lecture Sec. No. 42, Institute for Fluid Dynamics and Applied Mathematics, Univ. of Maryland, 1964.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198909779294
PII:
S 00029939(1989)09779294
Article copyright:
© Copyright 1989 American Mathematical Society
