The optimal weight of vibrating string equations with the first two eigenvalues based on critical equations

<jats:p> The present paper is concerned with the optimal weight of vibrating string equations with the first two eigenvalues <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0308210526101516_inline1.png"/> <jats:tex-math>$\lambda_1$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0308210526101516_inline2.png"/> <jats:tex-math>$\lambda_2$</jats:tex-math> </jats:alternatives> </jats:inline-formula> being given. Applying the method of critical equations in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0308210526101516_inline3.png"/> <jats:tex-math>$L^p[0,1]$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0308210526101516_inline4.png"/> <jats:tex-math>$p \gt 1$</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0308210526101516_inline5.png"/> <jats:tex-math>$\lambda_2 \ge 2\lambda_1$</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0308210526101516_inline6.png"/> <jats:tex-math>$L^1[0,1]$</jats:tex-math> </jats:alternatives> </jats:inline-formula> . </jats:p>