On majorants of eigenvalues of Sturm-Liouville problems with potentials from balls of weighted spaces

TL;DR

This paper proves the existence and uniqueness of potentials in certain weighted function spaces that maximize the minimal eigenvalue of Sturm-Liouville problems with Dirichlet boundary conditions.

Contribution

It establishes the existence and uniqueness of extremal potentials within weighted spaces for eigenvalue maximization in Sturm-Liouville problems.

Findings

Existence of a unique potential maximizing the minimal eigenvalue.

Extension of results to potentials in the closure of the class in generalized function spaces.

Identification of extremal potentials for specific weighted classes.

Abstract

It is constructively proved that for class $A_{r,\gamma}=\{q\in L_{1,loc}(0,1): q\leq 0, \int_0^1 rq^\gamma\,dx\leqslant 1\}$, where $r\in C[0,1]$ is uniformly positive weight and $\gamma>1$, there exists a unique potential $\hat q\in A_{r,\gamma}$ such that minimal eigenvalue $\lambda_0(\hat q)$ of boundary problem $$-y"+\hat qy=\lambda y, y(0)=y(1)=0 $$ is equal to $M_{r,\gamma}=\sup_{q\in A_{r,\gamma}}\lambda_0(q)$. For case $\gamma=1$ we obtain that there exists a unique potential $\hat q\in\Gamma_{r,\gamma}$ with analogous property. Here $\Gamma_{r,\gamma}$ is a closure of $A_{r,\gamma}$ in the space $W_{2,loc}^{-1}(0,1)$ of generalized functions.