Abstract:
Let $\Omega=\Omega_0\setminus\overline{\Theta}\subset\R^n$ ($n\ge 2$) such that $\Omega_0$ is open and convex and $\Theta$ is a finite union of open sets homeomorphic to balls.
We consider the eigenvalue problem for the Laplace operator associated to $\Omega$, with Robin boundary condition with parameter $\beta<0$ on $\partial \Omega_0$ and Neumann boundary condition on $\partial \Theta$. In 2020, G. Paoli, G. Piscitelli and L. Trani
proved that the spherical shell is the only maximizer for the first Robin-Neumann eigenvalue in the class of domains $\Omega$ with fixed volume and outer perimeter.
In this seminar we present the quantitative version of the afore-mentioned isoperimetric inequality and recall some previous stability results involving the Robin condition with
negative boundary parameter. The main novelty when dealing with sets of the type $\Omega=\Omega_0\setminus \overline{\Theta}$ consists in the introduction of a new type of hybrid asymmetry, that takes into account the different boundary conditions on $\partial\Omega_0$
and $\partial \Theta$. Up to our knowledge, in this context, this is the first stability result in which \emph{both} the outer and the inner boundary are perturbed.
The talk is based on some results obtained in collaboration with D.A. La Manna (Università degli Studi di Napoli Federico II), G. Paoli (Università degli Studi di Napoli Federico
II) and G. Piscitelli (Università degli Studi di Napoli Parthenope).