The present paper is concerned with the optimal weight of vibrating string equations with the first two eigenvalues ₁ and ₂ being given. Applying the method of critical equations in Lᵖ0, 1 for p 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 ₂ 2₁ 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 L¹0, 1.
Qi et al. (Fri,) studied this question.