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The optimal weight of vibrating string equations with the first two eigenvalues based on critical equations
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 24 April 2026, pp. 1-23
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The present paper is concerned with the optimal weight of vibrating string equations with the first two eigenvalues
$\lambda_1$ and 
$\lambda_2$ being given. Applying the method of critical equations in 
$L^p[0,1]$ for 
$p \gt 1$ and the inverse spectral theory of Sturm–Liouville problems with measure coefficients, we find that the optimal weight can be uniquely determined if and only if 
$\lambda_2 \ge 2\lambda_1$ provided that the weight is non-negative and symmetrical. As an application, we provide an estimation of the extremum for partial trace of the first two eigenvalues on a sphere in 
$L^1[0,1]$.
