Publication Date:
2019
Description:
〈p〉Publication date: 15 March 2019〈/p〉
〈p〉〈b〉Source:〈/b〉 Journal of Differential Equations, Volume 266, Issue 7〈/p〉
〈p〉Author(s): Xijun Hu, Lei Liu, Li Wu, Hao Zhu〈/p〉
〈h5〉Abstract〈/h5〉
〈div〉〈p〉It is natural to consider continuous dependence of the 〈em〉n〈/em〉-th eigenvalue on 〈em〉d〈/em〉-dimensional (〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"〉〈mi〉d〈/mi〉〈mo〉≥〈/mo〉〈mn〉2〈/mn〉〈/math〉) Sturm–Liouville problems after the results on 1-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all the boundary conditions such that the 〈em〉n〈/em〉-th eigenvalue is not continuous, and give complete characterization of asymptotic behavior of the 〈em〉n〈/em〉-th eigenvalue. This renders a precise description of the jump phenomena of the 〈em〉n〈/em〉-th eigenvalue near such a boundary condition. Furthermore, we divide the space of boundary conditions into 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"〉〈mn〉2〈/mn〉〈mi〉d〈/mi〉〈mo〉+〈/mo〉〈mn〉1〈/mn〉〈/math〉 layers and show that the 〈em〉n〈/em〉-th eigenvalue is continuously dependent on Sturm–Liouville equations and on boundary conditions when restricted into each layer. In addition, we prove that the analytic and geometric multiplicities of an eigenvalue are equal. Finally, we obtain derivative formula and positive direction of eigenvalues with respect to boundary conditions.〈/p〉〈/div〉
Print ISSN:
0022-0396
Electronic ISSN:
1090-2732
Topics:
Mathematics
