A lower bound for the principal eigenvalue of fully nonlinear elliptic operators

Abstract

<p style='text-indent:20px;'>In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We illustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \lim_{p\to \infty}\lambda_{1,p}(\Omega) = \lambda_{1,\infty}(\Omega) = \left(\frac{\pi}{2R}\right)^2 $\end{document} </tex-math> </disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ R $\end{document}</tex-math></inline-formula> is the largest radius of a ball included in the domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset {\mathbb R}^n $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M3">\begin{document}$ \lambda_{1,p}(\Omega) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \lambda_{1,\infty}(\Omega) $\end{document}</tex-math></inline-formula> are the principal eigenvalue for the homogeneous <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula>-laplacian and the homogeneous infinity laplacian respectively.