In this paper, we will first establish the necessary and sufficient conditions for the existence of the principal eigenvalues of second-order measure differential equations with indefinite weighted measures subject to the Neumann boundary condition. Then we will show the principal eigenvalues are continuously dependent on the weighted measures when the weak * topology is considered for measures. As applications, we will finally solve several optimization problems on principal eigenvalues, including some isospectral problems.

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We first use the Schwarz rearrangement to solve a minimization problem on eigenvalues of the one-dimensional p-Laplacian with integrable potentials. Then we construct an optimal class of non-degenerate potentials for the one-dimensional p-Laplacian with the Dirichlet boundary condition. Such a class of nondegenerate potentials is a generalization of many known classes of non-degenerate potentials and will be useful in many problems of nonlinear differential equations. Keywords p-Laplacian, eigenvalue, minimization problem, Schwarz rearrangement, non-degenerate potential, boundary value problem MSC(2010) 34L15, 49R05, 34L40, 49J15, 34B05 Citation: Wen Z Y, Zhang M R. Minimization of eigenvalues and construction of non-degenerate potentials for the one-dimensional p-Laplacian.

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