Optimal bounds for the first eigenvalue of fourth-order beam equations with integrable potentials

In this paper we study the optimization problems of the first eigenvalue for the fourth-order beam equations with integrable potentials, given the L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>1$$\end{document} norm of the potentials. We reduce these infinite-dimensional optimization problems of implicit functionals to finite-dimensional ones and derive optimal bounds for the first eigenvalue. Our results can be regarded as a contribution to the literature by developing a methodological framework for solving these infinite-dimensional optimization problems.