{"ID":2853175,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.16866","arxiv_id":"2510.16866","title":"One-dimensional optimisation of indefinite-weight principal eigenvalues under inhomogeneous Robin conditions and a Schrödinger-type perturbation","abstract":"We study the minimisation of the positive principal eigenvalue for an indefinite-weight problem under inhomogeneous Robin boundary conditions. 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 solve the one-dimensional optimisation problem completely: the optimal favourable set is an interval, and we determine whether it attaches to the left endpoint, to the right endpoint, or lies in the interior according to explicit criteria involving the two boundary parameters. The classification is then verified numerically by a transfer--matrix shooting method. 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 stability result showing that minimisers for small background potential converge to those of the unperturbed problem. This perturbed regime is likewise illustrated numerically.","short_abstract":"We study the minimisation of the positive principal eigenvalue for an indefinite-weight problem under inhomogeneous Robin boundary conditions. The model is motivated by diffusive logistic equations in spatially heterogeneous environments, where the weight describes allocatable favourable resources and the Robin paramet...","url_abs":"https://arxiv.org/abs/2510.16866","url_pdf":"https://arxiv.org/pdf/2510.16866v2","authors":"[\"Baruch Schneider\",\"Diana Schneiderová\",\"Yifan Zhang\"]","published":"2025-10-19T14:57:10Z","proceeding":"math.SP","tasks":"[\"math.SP\",\"math.OC\"]","methods":"[]","has_code":false}