Consider the following one-dimensional eigenvalue problem \begin{align*} -\frac{d}{dx}\left(\sigma(x)\frac{du}{dx}\right) & = \lambda u \ \ \textrm{ in $(0,L)$} \\ u(0) = u(L) & = 0, \end{align*} where the eigenvalue $\lambda = \lambda(\sigma)$ is a function of $\sigma$. I am interested in numerically finding the optimal $\sigma$ that maximises the first eigenvalue $\lambda_1(\sigma)$, i.e. solve the following spectral optimisation problem $$ \max_{\sigma\in\mathcal{A}} \lambda_1(\sigma), $$ where $\mathcal{A}$ is the set of admissible functions, given by $$ \mathcal{A} = \left\{\sigma\in L^\infty(0,L)\colon 0<\alpha\le\sigma(x)\le\beta \ \ a.e.,\, \int_0^L \sigma\, dx = C\right\}, $$ for some fixed positive constants $\alpha,\beta, C$. I am hoping that someone could point me to some references/articles (if any)! Feel free to edit the title of this question and the tag as you see fit.
Note: I do know that the exact solution can be found using Krein's theorem, and the optimal solution $\sigma_c$ is unique and of bang-bang type, i.e. $$ \sigma_c = \begin{cases} \, \beta & \ \ \textrm{ for $x\in(0,L/2 - \delta)$}, \\ \, \alpha & \ \ \textrm{ for $x\in(L/2 - \delta,L/2 + \delta)$}, \\ \, \beta & \ \ \textrm{ for $x\in(L/2 + \delta,L)$}, \end{cases} $$ where $\delta$ is $\dfrac{\beta L - C}{2(\beta - \alpha)}$.