Extremal Values of Half-Eigenvalues for -Laplacian with Weights in Balls
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Research Article Extremal Values of Half-Eigenvalues for p-Laplacian with Weights in L1 Balls Ping Yan Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China Correspondence should be addressed to Ping Yan, [email protected] Received 24 May 2010; Accepted 21 October 2010 Academic Editor: V. Shakhmurov Copyright q 2010 Ping Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For one-dimensional p-Laplacian with weights in Lγ : Lγ 0, 1, R 1 ≤ γ ≤ ∞ balls, we are interested in the extremal values of the mth positive half-eigenvalues associated with Dirichlet, Neumann, and generalized periodic boundary conditions, respectively. It will be shown that the extremal value problems for half-eigenvalues are equivalent to those for eigenvalues, and all these extremal values are given by some best Sobolev constants.
1. Introduction Occasionally, we need to solve extremal value problems for eigenvalues. A classical example studied by Krein 1 is the infimum and the supremum of the mth Dirichlet eigenvalues of Hill’s operator with positive weight inf μD m w : w ∈ Er,h ,
sup μD m w : w ∈ Er,h ,
1.1
where 0 < r ≤ h < ∞ and Er,h :
w ∈ L : 0 ≤ w ≤ h, γ
1
wtdt r .
1.2
0
In this paper, we always use superscripts D, N, P , and G to indicate Dirichlet, Neumann, periodic and generalized periodic boundary value conditions, respectively. Similar extremal value problems for p-Laplacian were studied by Yan and Zhang 2. For Hill’s operator with weight, Lou and Yanagida 3 studied the minimization problem of the positive principal
2
Boundary Value Problems
Neumann eigenvalues, which plays a crucial role in population dynamics. Given constants κ ∈ 0, ∞ and α ∈ 0, 1, denote Sκ,α :
∞
ω ∈ L : −1 ≤ ω ≤ κ, ω 0,
1
ωtdt ≤ −α .
1.3
0
The positive principal eigenvalue μN 0 ω is well-defined for any ω ∈ Sκ,α , and the minimization problem in 3 is to find inf μN 0 w : ω ∈ Sκ,α .
1.4
In solving the previous three problems, two crucial steps have been employed. The first step is to prove that the extremal values can be attained by some weights. For regular self-adjoint linear Sturm-Liouville problems the continuous dependence of eigenvalues on weights/potentials in the usual Lγ topology is well understood, and so is the Fr´echet differentiable dependence. Many of these results are summarized in 4. It is remarkable that this step cannot be answered immediately by such a continuity results, because the space of weights is infinite-dimensional. The second step is to find the minimizers/maximizers. This step is tricky and it depends on the problem studied. For L1 weights the solution is suggested by the Pontrjagin’s Maximum Principle 5, Sections 48.6–48.8. For Sturm-Liouville operators and Hill’s operators Zhang 6 proved that the eigenvalues are continuous in potentials in
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