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Eigenvalue Ratios for vibrating string equations with single-well densities

Jihed Hedhly

Abstract

In this paper, we prove the optimal upper bound $\frac{λ_n}{λ_m}\leq(\frac{n}{m})^2$ of vibrating string $$-y''=λρ(x) y,$$ with Dirichlet boundary conditions for single-well densities. The proof is based on the inequality $\frac{λ_n(ρ)}{λ_{m}(ρ)}\leq \frac{λ_n(L)}{λ_{m}(L)} ,$ with $L$ must be a stepfunction. We also prove the same result for the Dirichlet Sturm-Liouville problems.

Eigenvalue Ratios for vibrating string equations with single-well densities

Abstract

In this paper, we prove the optimal upper bound of vibrating string with Dirichlet boundary conditions for single-well densities. The proof is based on the inequality with must be a stepfunction. We also prove the same result for the Dirichlet Sturm-Liouville problems.

Paper Structure

This paper contains 4 sections, 8 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Eigenvalue ratio for the vibrating string equations
  3. Proof of Theorems \ref{['theo11']}
  4. Eigenvalue ratios for Sturm-Liouville problems

Key Result

Proposition 1

Let $\rho>0$ be monotone decreasing in $[0,1]$ and let $L(x)=\rho(x_{2i+1})$ (where $x_i$ the points such that $u_n ^ 2 (x_i) = u_ {n-1}^2 (x_i) )$, then with equality if and only if $\rho\equiv L$.

Theorems & Definitions (9)

  • Proposition 1
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • Corollary 2
  • Theorem 2
  • Corollary 3
  • Lemma 2
  • Remark 1