Minimization of eigenvalues and construction of non-degenerate potentials for the one-dimensional p-Laplacian

Abstract

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.

A variational approach to the optimal locations of the nodes of the second Dirichlet eigenfunctions

“…It is worth mentioning that, after the obtention of CC and CD of eigenvalues and eigenfunctions in potentials and in weights, the direct variational method has been applied to solve a series of optimization problems on eigenvalues of different types of differential equations. Besides the early papers of Wei et al 10 and Zhang 11 of ours, see also previous works 19–32 . In fact, such an approach to the optimization problems on eigenvalues is relatively new and is very fruitful.…”

“…Besides the early papers of Wei et al 10 and Zhang 11 of ours, see also previous works. [19][20][21][22][23][24][25][26][27][28][29][30][31][32] In fact, such an approach to the optimization problems on eigenvalues is relatively new and is very fruitful. Besides, the answers to these optimization problems also provide answers to their inverse problems, like (1.11).…”

Minimization of first eigenvalue of fourth order measure differential equation

“…上述引理的结论告诉我们,L(r) 与 L(r) 有相同的表达式. 此外, 本文所解决的极小值问题可以 应用到与四阶测度微分方程相关的其他问题中, 如非退化势的构造 [17] 、建立 Lyapunov 不等式 [18] 等 问题.…”