Principal eigenvalue of a very badly degenerate operator and applications
The author considers the eigenvalue problem \[ -\Delta_{\infty}u=\lambda u\;{\text{in}} \;\Omega,\qquad u=0\;{\text{on}} \;\partial\Omega\,, \] where \(\Omega\) is a bounded open subset of \(\mathbb R^{n}\), \(u\) is a function of \(\Omega\) to \(\mathbb R\), \(\lambda\) plays the role of an eigenvalue, and \[ \Delta_{\infty}u\equiv \biggl(D^{2}u \frac{Du}{| Du| }\biggr) \frac{Du}{| Du| }. \] It is proved that the least eigenvalue is positive and is delivered by the formula \[ \lambda_{1}=\sup\{\lambda:\exists v>0\text{ in }\Omega,\text{ such that } -\Delta_{\infty}v\geq\lambda v\}. \] The author also proves that \(\lambda_{1}\) admits a positive eigenfunction and that it can be characterized as the supremum of the values \(\lambda\) for which \(\Delta_{\infty}+\lambda I\) satisfies the maximum principle. Then some applications are presented to the analysis of the problem \[ -\Delta_{\infty}\varphi= \lambda\varphi +f \quad \text{in }\Omega, \] with respect to unknown \(\varphi\) for some positive function \(f\) of \(\Omega\) to \(\mathbb R\).
scientific article; zbMATH DE number 5161141
scientific article