Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight
Motivated by extremal problems of weighted Dirichlet or Neumann eigenvalues, we will establish two fundamental results on the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights. One is the continuous differentiability of eigenvalues in weights in the Lebesgue spaces L γ with the usual norms. Another is the continuity of eigenvalues in weights with respect to the weak topologies in L γ spaces. Here 1 ≤ γ ≤ ∞. In doing so, we will give a simpler explanation to the corresponding spectrum problems, with the help of several typical techniques in nonlinear analysis such as the Fréchet derivative and weak* convergence.
Mathematics Subject Classification (2010)Primary 58C07 47J10 Secondary 34L15 58C40
KeywordsSpectrum eigenvalue p-Laplacian weight continuity weak topology differentiability
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- 1.Anane A., Chakrone O., Monssa M. (2002) Spectrum of one dimensional p- Laplacian with indefinite weight. Electr. J. Qualitative Theory Differential Equations 2002(17): 11Google Scholar
- 11.Krein M.G. (1955) On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability. Amer. Math. Soc. Transl. Ser. 2(1): 163–187Google Scholar
- 13.Megginson R.E. (1998) An Introduction to Banach Space Theory, Graduate Texts Math., Vol. 183. Springer-Verlag, New YorkGoogle Scholar
- 14.G. Meng and M. Zhang, Continuity in weak topology: first order linear systems of ODE, Preprint, 2008. http://faculty.math.tsinghua.edu.cn/~mzhang/
- 16.Pöschel J., Trubowitz E. (1987) The Inverse Spectral Theory. Academic Press, New YorkGoogle Scholar
- 19.P. Yan and M. Zhang, Continuity in weak topology and extremal problems of eigenvalues of the p-Laplacian, Trans. Amer. Math. Soc., in press.Google Scholar