Extremum estimates of the \(L^1\)-norm of weights for eigenvalue problems of vibrating string equations based on critical equations (Q2033743)

The authors consider the boundary value problem \begin{align*} -y''(x)=\lambda w(x)y(x),\tag{1}\\ y(x)=0=y(1)-hy'(1), \quad 0<h<1\tag{2}, \end{align*} where \(0\ne \lambda\in \mathbb R\), \(w\in L^1[0,1]\) and \(mes \{x:w(x)>0, \, x\in [0,1]\}>0\) as well as \(mes \{x:w(x)<0, \, x\in [0,1]\}>0\). The expression of the infimum function \(E(\lambda, h)\) of \(||w||_1\) is given if (1) and (2) has nontrivial solution i.e \(\int_0^1|w|\ge E(\lambda,h)\) and \[ E(\lambda, h)=\inf\{||w||_1: w\in \Omega(\lambda)\},\tag{3} \] where \[ \Omega(\lambda)=\left\{w\in L^1[a,b]: \lambda\in \sigma(w)\cap\mathbb R, \int_0^1|w|>0\right\}. \] The authors apply the critical equation weights to solve the extremal problem (3). They obtain the following main result: ``If the problem (1) and (2) has nontrivial solution for \(0\ne \lambda\in \mathbb R\), \(w\in L^1[0,1]\) and \(h\in(0,1)\), then \[ E(\lambda, h)=\dfrac1{|\lambda|}\in\left\{\dfrac{1-h}h, \dfrac4{1-h}\right\}. \]