One-dimensional optimisation of indefinite-weight principal eigenvalues with asymmetric Robin parameters and a Schr\"odinger-type perturbation

We study the minimisation of the positive principal eigenvalue for an indefinite-weight problem with asymmetric Robin parameters.

The model is motivated by diffusive logistic equations in spatially heterogeneous environments, where the weight describes allocatable favourable resources and the Robin parameters measure boundary loss.

After recalling the variational setting and the bang-bang reduction, we analyse the one-dimensional optimisation problem: the optimal favourable set is an interval, and the placement problem is reduced to a branchwise criterion.

The key analytical tool is a shape-derivative formula for $a\mapsto\lambda(a)$, which shows that interior candidates are characterised by equality of the endpoint values of the positive eigenfunction, equivalently by the coupled transfer-matrix equations $f=0$ and $g=0$.

We also introduce a Schrödinger-type extension with a fixed nonnegative background potential.

In the coercive case we establish the corresponding principal-eigenvalue and bang-bang results, and in one dimension with constant potential we prove a compactness-type stability result showing that minimisers for small background potential converge, along subsequences, to minimisers of the unperturbed problem.

No placement classification is claimed for general positive background potential.

The computations are presented as numerical illustrations generated with an adaptive root-search protocol.