Extremal Problems of the Density for Vibrating String Equations with Applications to Gap and Ratio of Eigenvalues

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).…”

The eigenvalue ratio of the vibrating strings with mixed boundary condition

“…Huang [17] provided the upper bounds on the eigenvalue ratio for vibrating strings with Dirichlet condition when V(x) satisfies certain conditions, and investigated the nature of the eigenvalues for vibrating strings [18]. Qi et al [19] obtained the infimum of the densities for vibrating string equations in terms of the gap and ratio of the first two eigenvalues. Hedhly [20] proved that the optimal upper bound…”

“…Huang [17] provided the upper bounds on the eigenvalue ratio for vibrating strings with Dirichlet condition when $$$ V(x) $$$ satisfies certain conditions, and investigated the nature of the eigenvalues for vibrating strings [18]. Qi et al [19] obtained the infimum of the densities for vibrating string equations in terms of the gap and ratio of the first two eigenvalues. Hedhly [20] proved that the optimal upper bound $$$ \frac{\lambda_n}{\lambda_m}\le {\left(\frac{n}{m}\right)}^2 $$$ of vibrating string equations under Dirichlet boundary condition for single‐well densities and the equality holds if and only if the density is a constant.…”

The fundamental gap of a kind of sub-elliptic operator

“…In particular, Laplacian fundamental gap problem [39,6,18,12] has been extended to different complex spaces and the corresponding detailed characterization is given. The lower bound of eigenvalue gap of vibrating string is also discussed in [22,8,37]. For the latest references, we can refer to [27,28,2].…”