×

Second order elliptic equations with degenerate weight. (English) Zbl 0703.35134

The author considers the eigenvalue problem \[ (1)\quad \ell u\equiv - \Delta u-qu=\lambda \omega u\text{ for } u\in \overset \circ H^{1,2}(\Omega). \] Here \(\Omega \subset {\mathbb{R}}^ n\) stands for a smooth bounded domain and \(q,\omega \in L^{\infty}(\Omega)\) are given functions such that (i) \(\omega\equiv 0\) a.e. in a subdomain \(\Omega ''\subset \Omega\), (ii) in the complementary (nonempty) subdomain \(\Omega '\) defined by \(\Omega\)-\({\bar \Omega}{}'=\Omega ''\) one has \(\omega\geq 0\) a.e., and (iii) there exists a \(\phi \in C^{\infty}_ 0(\Omega)\) satisfying \((\ell \phi,\phi)<0.\)
Then he shows that:
(1) The problem (1) has infinitely many real eigenvalues,
(2) The standard Courant min-max principle may not be valied, and
(3) The number of negative eigenvalues depend on \(\Omega\), \(\Omega '\) and \(\Omega ''\) but not on the specific values of \(\omega\) on \(\Omega '.\)
He gives also both an exact formula for the number of negative eigenvalues and a suitable modification of the Courant principle. The main results are stated in the form of five theorems. They are applied to estimate the location of the negative spectrum of \(\ell\), if any.
Remark that the case \(n=1\) (the case of ordinary differential equations) was previously studied by others. Remark also that the results seem to be extendible with very little changes to both more general uniformly elliptic expressions in place of \(\Delta\) and more general boundary conditions by assuming \(u\in V\), with \(\overset \circ H^{1,2}(\Omega)\subset V\subset H^{1,2}(\Omega)\), in place of \(u\in \overset \circ H^{1,2}(\Omega)\).
Reviewer: M.Idemen

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J20 Variational methods for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Walter Allegretto and Angelo B. 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), no. 1, 69 – 73. · Zbl 0585.35026
[2] -, Boundary problems of the second order with an indefinite weight function, preprint.
[3] Paul Binding and Patrick J. Browne, Spectral properties of two-parameter eigenvalue problems. II, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 39 – 51. · Zbl 0642.47003 · doi:10.1017/S0308210500018187
[4] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. · Zbl 0051.28802
[5] W. N. Everitt, Man Kam Kwong, and A. Zettl, Oscillation of eigenfunctions of weighted regular Sturm-Liouville problems, J. London Math. Soc. (2) 27 (1983), no. 1, 106 – 120. · Zbl 0529.34039 · doi:10.1112/jlms/s2-27.1.106
[6] W. N. Everitt, Man Kam Kwong, and A. Zettl, Differential operators and quadratic inequalities with a degenerate weight, J. Math. Anal. Appl. 98 (1984), no. 2, 378 – 399. · Zbl 0559.34021 · doi:10.1016/0022-247X(84)90256-7
[7] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[8] I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. · Zbl 0194.43804
[9] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. · Zbl 0063.02971
[10] Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0242.46001
[11] Franz Rellich, Perturbation theory of eigenvalue problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon and Breach Science Publishers, New York-London-Paris, 1969. · Zbl 0181.42002
[12] Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. · Zbl 0070.10902
[13] S. L. Sobolev, Applications of functional analysis in mathematical physics, Translated from the Russian by F. E. Browder. Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963. · Zbl 0123.09003
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.