Sharp bounds of eigenvalues and related inverse problems for Euler-Bernoulli beam equations

In this paper, we develop an analytical method to prove the sharp upper and lower bounds for the first eigenvalue of the so-called Euler-Bernoulli beam equations, provided the ℒ 1 -norms and the average of the potentials. In particular, we focus on the indefinite potentials. We also study a kind of inverse spectral problems, which means that the optimal lower bounds of ℒ 1 -norms of the potentials can be obtained when the first eigenvalue and the average of the potentials are fixed. We use the setting of measure differential equations to understand the direct spectral problems as well as the inverse problems because the solutions for them will lead to Dirac measure distributions of potentials.