A sharp bound for the first Robin-Dirichlet eigenvalue (Q6636809)

The authors prove that the lowest eigenvalue of the Laplace operator in a doubly connected domain \(\Omega\) of \(\mathbb R^n\), with mixed Robin and Dirichlet boundary conditions, is maximized by the lowest eigenvalue in a spherical shell \(A_{R_1,R_2}:=\{\mathrm x\in\mathbb R^n\colon R_1<|\mathrm x|<R_2\}\), provided that\N\begin{itemize}\N\item \(\Omega=\Omega_0\setminus\overline{\Theta}\), where \(\Omega_0\) and \(\Theta\) are both bounded, open and convex, and \(\overline{\Theta}\subset \Omega_0\);\N\item The Dirichlet boundary condition is imposed on the inner boundary, \(\partial\Theta\), and the Robin parameter has positive sign;\N\item \(\Omega\) obeys a constraint defined by the measure of \(\Omega\), the surface measure of the outer boundary of \(\Omega\), and the \((n-1)\)th quermassintegral of the hole \(\Theta\);\N\item \(A_{R_1,R_2}\) and \(\Omega\) have equal measures, \(B_{R_2}\) and \(\Omega_0\) have equal perimeters, and \(B_{R_1}\) and \(\Theta\) have equal \((n-1)\)th quermassintegrals.\N\end{itemize}