Pólya-type estimates for the first Robin eigenvalue of elliptic operators

• Published in: arXiv.org
• Main Author: F Della Pietra
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 13.02.2024
• Subjects: Dirichlet problem; Eigenvalues; Estimates; Lower bounds; Real numbers; Upper bounds
• ISSN: 2331-8422

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic \(p\)-Laplace operator, namely: \[ \lambda_F(\beta,\Omega)=\lambda_{F}(p,\beta,\Omega)= \min_{\psi\in W^{1,p}(\Omega)\setminus\{0\} } \frac{\int_\Omega F(\nabla \psi)^p dx +\beta\int_{\partial\Omega}|\psi|^p F(\nu_{\Omega}) d\mathcal H^{N-1} }{\int_\Omega|\psi|^p dx} \] where \(p\in]1,+\infty[\), \(\Omega\) is a bounded, convex domain in \(\mathbb R^{N}\), \(\nu_{\Omega}\) is its Euclidean outward normal, \(\beta\) is a real number, and \(F\) is a sufficiently smooth norm on \(\mathbb R^{N}\). We show an upper bound for \(\lambda_{F}(\beta,\Omega)\) in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on \(\beta\) and on the volume and the anisotropic perimeter of \(\Omega\), in the spirit of the classical estimates of Pólya \cite{po61} for the Euclidean Dirichlet Laplacian. We will also provide a lower bound for the torsional rigidity \[ \tau_p(\beta,\Omega)^{p-1} = \max_{\substack{\psi\in W^{1,p}(\Omega)\setminus\{0\}}} \dfrac{\left(\int_\Omega |\psi| \, dx\right)^p}{\int_\Omega F(\nabla\psi)^p dx+\beta \int_{\partial\Omega}|\psi|^p F(\nu_{\Omega}) d\mathcal H^{N-1} }, \] when \(\beta>0\). The obtained results are new also in the case of the classical Euclidean Laplacian.