发表机构
University of Ostrava; Charles University; VŠB–Technical University of Ostrava（俄斯特拉发大学; 查理大学; 俄斯特拉发工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对Robin边界条件下带不定权重的椭圆特征值问题，确定了小有利区域的最优位置，揭示了边界与内部构型转变的阈值，并给出了一阶选择定律及数值验证。
AI 中文摘要
我们考虑具有Robin边界条件和bang-bang不定权重$\kappa\mathbf 1_D-\mathbf 1_{\Omega\setminus D}$的椭圆问题的正主特征值，并询问体积为$|D|=\delta$的给定小体积的有利区域$D$应位于何处。设$\varepsilon=\delta^{1/N}$和$\tau_\delta=\alpha_\delta\varepsilon$。我们证明存在一个有限阈值$\tau_*=\tau_*(N,\kappa)$，独立于环境域，使得当$\tau_\delta\to\tau<\infty$时，$\delta^{2/N}\Lambda_\delta(\alpha_\delta)\to\Lambda_{\mathbb H}(\tau)$。如果$\tau<\tau_*$，最优小区域集中在边界上。如果$\tau>\tau_*$，它们的集中中心移动到距边界远大于$\varepsilon$的距离，并且在重新中心和重新缩放后，有利集在测度上收敛到全空间最优球。在$\tau=\tau_*$时，半空间问题允许一个紧致边界优化器以及逃逸到无穷远的极小化序列。在更精细的机制$\tau_\delta=\tau_*+\sigma\varepsilon+o(\varepsilon)$中，我们确定了边界和内部构型之间的一阶竞争。紧致阈值优化器的切向对称性将几何修正简化为平均曲率。特别地，每个固定的有限Robin系数渐近地处于边界机制中。数值实验说明了边界-内部转变、一阶选择定律中的三种情况以及依赖于曲率的边界位置；对于$N=2$和$\kappa=1$，它们表明转变发生在$\tau_*\approx3.2$附近。
英文摘要
We consider the positive principal eigenvalue of an elliptic problem with a Robin boundary condition and bang--bang indefinite weight $κ\mathbf 1_D-\mathbf 1_{Ω\setminus D}$, and ask where a favourable region $D$ of prescribed small volume $|D|=δ$ should be located. Put $\varepsilon=δ^{1/N}$ and $τ_δ=α_δ\varepsilon$. We prove that there is a finite threshold $τ_*=τ_*(N,κ)$, independent of the ambient domain, such that $δ^{2/N}Λ_δ(α_δ)\toΛ_{\mathbb H}(τ)$ whenever $τ_δ\toτ<\infty$. If $τ<τ_*$, optimal small regions concentrate at the boundary. If $τ>τ_*$, their concentration centres move to distances much larger than $\varepsilon$ from the boundary, and after recentring and rescaling the favourable sets converge in measure to the whole-space optimal ball. At $τ=τ_*$, the half-space problem admits a compact boundary optimiser as well as minimising sequences escaping to infinity. In the finer regime $τ_δ=τ_*+σ\varepsilon+o(\varepsilon)$, 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 $κ=1$ they suggest a transition near $τ_*\approx3.2$.
