Mediterranean Journal of Mathematics

, Volume 7, Issue 2, pp 225–248 | Cite as

Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight

Article

Abstract

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 

Keywords

Spectrum eigenvalue p-Laplacian weight continuity weak topology differentiability 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.Zhou Pei-Yuan Center for Applied MathematicsTsinghua UniversityBeijingChina

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