×

zbMATH — the first resource for mathematics

Strongly singular Sturm-Liouville problems. (English) Zbl 0994.34013
In the usual Sturm-Liouville theory, in order to use the spectral theorem for selfadjoint operators in Hilbert spaces, appropriate boundary conditions are used together with the Sturm-Liouville equation \[ -(ru'(x))'+ p(x) u(x)=\lambda m(x) u(x)\quad\text{on }(a,b) \] to form selfadjoint eigenvalue problems. For example, for the Fourier equation \(-u''=\lambda u\) on \((0,+\infty)\), such a boundary condition has the form \(u(0)\cos\alpha+ u'(0)\sin\alpha= 0\) with \(\alpha\in [0,\pi)\), since the endpoint \(+\infty\) is of the limit-point type.
Thus, for a singular Sturm-Liouville equation, i.e., when one of \(1/r\), \(p\) and \(m\) is not Lebesgue integrable on \((a,b)\), some very “natural” eigenvalue problems are nonselfadjoint. For example, the problem consists of the Fourier equation on \((0,+\infty)\) and the periodic boundary condition, i.e., \(u(0)= u(+\infty)\) and \(u'(0)= u'(+\infty)\), is nonselfadjoint. The eigenvalues of such problems are complicated in general. For example, \(0\) is the only eigenvalue of the above concrete problem. The author deals with this type of eigenvalue problems.
Here, \(r: (a,b)\to \mathbb{R}^+\) is continuous, \(p,m\in L^1_{\text{loc}}((a,b),\mathbb{R})\), \(p\geq 0\), and there is a \(c\in (a,b)\) such that \(1/r\in L^1(a,c)\), \(1/r\not\in L^1(c,b)\), \(pr_1\not\in L^1(a,c)\), and \(pr_1\not\in L^1(c,b)\), where \(r_1(t)= \int^t_a ds/r(s)\). The eigenvalue problem considered here consists of a Sturm-Liouville equation satisfying these assumptions and the Dirichlet boundary condition \(u(a)= u(b)= 0\), and is said to be strongly singular.
The main results concerning the eigenvalues can be stated as follows: If \(m\in L^1(a,c)\) and \(mr_1\in L^1(c,b)\) for some \(c\in (a,b)\), then all eigenvalues are simple, and together with \(0\) they can be ordered to form a sequence \[ \cdots<\lambda_{-2}< \lambda_{-1}< 0< \lambda_1< \lambda_2<\cdots\;. \] Each of the sequences \(\dots,\lambda_{-2},\lambda_{-1}\) and \(\lambda_1,\lambda_2,\dots\) can be void, finite, or infinite with infinity as its limit. If, in addition, \(mr^{3/2}_1\in L^1(c,b)\) for some \(c\in (a,b)\) and \(\{t\in (a,b); m(t)> 0\}\) has a positive measure, then \(\lambda_1\) exists and is given by \[ \min\left\{ \int^b_a (ru^{\prime 2}+ pu^2)\Biggl/ \int^b_a mu^2;\;\begin{matrix} u\in C[a,b),\;u(a)= 0\\ \sqrt pu,\sqrt ru'\in L^2(a,b), \int^b_a mu^2> 0\end{matrix}\right\}. \] A maximum principle and an anti-maximum principle concerning \(\lambda_1\) are proved. Moreover, the author also gives examples to show that under the assumption in this paper, all four possibilities of the limit-point and limit-circle classification occur.

MSC:
34B24 Sturm-Liouville theory
34B27 Green’s functions for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amann, SIAM Rev. 18 pp 620– (1976)
[2] : A Topological Introduction to Nonlinear Analysis, Birkhauser, Boston, 1993
[3] Cecchi, J. Diff. Equations 82 pp 15– (1989)
[4] Cecchi, J. Diff. Equations 99 pp 381– (1992)
[5] Cecchi, J. Diff. Equations 118 pp 403– (1995)
[6] Duhoux, Proc. Roy. Soc. Edinburgh 128A pp 525– (1998) · Zbl 0983.34015 · doi:10.1017/S0308210500021636
[7] Duhoux, Nonlinear Anal. 38 pp 897– (1999)
[8] Duhoux, Tatra Mt. Math. Publ. 19 pp 1– (2000)
[9] Hajmirzaahmad, SIAM Rev. 34 pp 614– (1992)
[10] and : Singular Self-Adjoint Sturm-Liouville Problems, Differential and Integral Equations 1 (4) (1988), 423-432 · Zbl 0723.34023
[11] Marini, J. Diff. Equations 28 pp 1– (1978)
[12] : Métodos da Teoria de Pontos Críticos, Textos de Matemática, Universidade de Lisboa, Faculdade de Ciências, Departamento de Matemática, 1993
[13] : Topological Vector Spaces, Springer-Verlag, New York, 1971.
[14] : Spectral Theory of Ordinary Differential Operators, Lectures Notes in Mathematics 1258, Springer-Verlag, New York, 1987
[15] : Analyse Convexe et Optimisation, CIACO, Louvain-la-Neuve, 1989
[16] : Analyse Harmonique Réelle, Hermann, Paris, 1995
[17] : Sturm-Liouville Problems. In: Spectral Theory and Computational Methods of Sturm-Liouville Problems, edited by D. Hinton and P. W. Schaefer, pp. 1-104, Marcel Dekker, New York, 1997
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.