Spectral optimization for the Stekloff-Laplacian: The stability issue (Q425742)

In the paper some spectral optimization problems are considered; in particular, the authors prove the following quantitative sharp inequality NEWLINE\[NEWLINE|\Omega|^{1/N}\sigma_2(\Omega)\geq |B|^{1/N}\sigma_2(B) [1- \alpha_N{\mathcal A}(\Omega)^2],NEWLINE\]NEWLINE where \(\sigma_2(\Omega)\) is the second Stekloff eigenvalue NEWLINE\[NEWLINE\sigma_2(\Omega)= \text{inf}\Biggl\{{\int_\Omega|\nabla u|^2 dx\over \int_{\partial\Omega} u^2d{\mathcal H}^{N-1}}: \int_{\partial\Omega} ud{\mathcal H}^{N-1}= 0\Biggr\},NEWLINE\]NEWLINE \(\alpha_N\) is a constant depending only on the dimension \(N\), and \({\mathcal A}(\Omega)\) is the Fraenkel asymmetry NEWLINE\[NEWLINE{\mathcal A}(\Omega)= \text{inf}\Biggl\{{|\Omega_\Delta B|\over |\Omega|}: B\text{ ball},\,|B|= |\Omega|\Biggr\}.NEWLINE\]NEWLINE A crucial tool in the proof consists in the fact that, for a given volume, the ball minimizes the weighted perimeter NEWLINE\[NEWLINE\int_{\partial\Omega} |x|^2\,d{\mathcal H}^{N-1},NEWLINE\]NEWLINE for which the authors prove the quantitative inequality NEWLINE\[NEWLINE\int_{\partial\Omega} |x|^2d{\mathcal H}^{N-1}\geq \int_{\partial\Omega} |x|^2d{\mathcal H}^{N-1} \Biggl[1+ \beta_N\Biggl({|\Omega_\Delta B|\over |\Omega|}\Biggr)^2\Biggr]NEWLINE\]NEWLINE for a suitable constant \(\beta_N\). Similar inequalities are also proved for the weight functions \(V(|x|)= |x|^p\) with \(p>1\).