Optimal location of small favourable regions for Robin eigenvalues with indefinite weights

Abstract

We consider the positive principal eigenvalue of an elliptic problem with a Robin boundary condition and bang--bang indefinite weight $\kappa 1_D-1_{\Omega \setminus D}$, and ask where a favourable region $D$ of prescribed small volume $|D|=\delta$ should be located. Put $\epsilon ={\delta }^{1/N}$ and ${\tau }_{\delta }={\alpha }_{\delta }\epsilon$. We prove that there is a finite threshold ${\tau }_{\ast }={\tau }_{\ast }(N,\kappa )$, independent of the ambient domain, such that ${\delta }^{2/N}{\Lambda }_{\delta }({\alpha }_{\delta })\to {\Lambda }_{\mathbb{H}}(\tau )$ whenever ${\tau }_{\delta }\to \tau <\infty$. If $\tau <{\tau }_{\ast }$, optimal small regions concentrate at the boundary. If $\tau >{\tau }_{\ast }$, their concentration centres move to distances much larger than $\epsilon$ from the boundary, and after recentring and rescaling the favourable sets converge in measure to the whole-space optimal ball. At $\tau ={\tau }_{\ast }$, the half-space problem admits a compact boundary optimiser as well as minimising sequences escaping to infinity. In the finer regime ${\tau }_{\delta }={\tau }_{\ast }+\sigma \epsilon +o(\epsilon )$, we determine the first-order competition between the boundary and interior configurations. Tangential symmetry of compact threshold optimisers reduces the geometric correction to mean curvature. In particular, every fixed finite Robin coefficient is asymptotically in the boundary regime. Numerical experiments illustrate the boundary--interior transition, the three cases in the first-order selection law, and the curvature-dependent boundary location; for $N=2$ and $\kappa =1$ they suggest a transition near ${\tau }_{\ast }\approx 3.2$.