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Characterization of Maximizers for Sums of the First Two Eigenvalues of Sturm-Liouville Operators

Gang Meng, Yuzhou Tian, Bing Xie, Meirong Zhang

Abstract

In this paper we study the maximization of the sum of the first two Dirichlet eigenvalues for Sturm-Liouville operators with potentials in the noncompact space $L^1$. We prove that there exists a unique potential function achieving the maximum, which is non-negative, piecewise smooth, and symmetric. Using measure differential equations and weak$^*$ convergence, we show that the nonzero part of the maximizer can be determined by the solution to the pendulum equation $θ'' + \ell \sinθ= 0 $.

Characterization of Maximizers for Sums of the First Two Eigenvalues of Sturm-Liouville Operators

Abstract

In this paper we study the maximization of the sum of the first two Dirichlet eigenvalues for Sturm-Liouville operators with potentials in the noncompact space . We prove that there exists a unique potential function achieving the maximum, which is non-negative, piecewise smooth, and symmetric. Using measure differential equations and weak convergence, we show that the nonzero part of the maximizer can be determined by the solution to the pendulum equation .
Paper Structure (5 sections, 12 theorems, 100 equations, 4 figures)

This paper contains 5 sections, 12 theorems, 100 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Measures and measure differential equations
  4. The critical systems of the optimization problem
  5. Main results

Key Result

Theorem 1.1

For any $r>0$, there exists a unique potential $\check{q}_r\in S_1[r]$ such that the maximum in (Mr1) is attained by $\check{q}_r$, i.e. Meanwhile, this potential $\check{q}_r$ is non-positive, piecewise smooth, and symmetric. Moreover, there exists a constant $\check c$ such that the nonzero part of $\check{q}_r$ is given by where the length $\ell$ is and $\theta=\theta(t) \in (-\pi, \pi)$ is

Figures (4)

  • Figure 1: The maximizing potential $q_p(t)=q_{p,r}(t)$ with $r=5$ and $p=15/14$.
  • Figure 2: The maximizing potential $\check{q}_{r}(t)$ (left) and eigenfunctions (right) with $r=5.$
  • Figure 3: The maximizing potential $\check{q}_{r}(t)$ (left) and eigenfunctions (right) with $r=20.$
  • Figure 4: The maximizing potential $\check{q}_{r}(t)$ (left) and eigenfunctions (right) with $r=r^*.$

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Lemma 3.4
  • ...and 6 more