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

This work has been partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina).
• 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

$\lim_{p\to \infty}\lambda_{1,p}(\Omega) = \lambda_{1,\infty}(\Omega) = \left(\frac{\pi}{2R}\right)^2$

where $R$ is the largest radius of a ball included in the domain $\Omega\subset {\mathbb R}^n$, and $\lambda_{1,p}(\Omega)$ and $\lambda_{1,\infty}(\Omega)$ are the principal eigenvalue for the homogeneous $p$-laplacian and the homogeneous infinity laplacian respectively.

Mathematics Subject Classification: Primary: 35P15; Secondary: 35P30, 35J60, 35J70.

 Citation: • • Figure 1.  Radial functions that allow us to obtain bounds for the principal eigenvalue

Figure 2.  In black an L shaped domain, in red the ball of maximum radius contained in the domain, in green the ball of minimum radius that contains the domain and in blue the boundary of the narrowest strip that contains the domain

Figure 3.  Functions $v$ (blue) and $\phi_{y_0}$ (red) defined in the proof of Theorem 2.2 for a square

Figure 4.  In black a U shaped domain ($\Omega$), in red the ball of maximum radius ($R$) included in $\Omega$, in blue $\Omega_\delta$ and in green the ball of maximum radius ($R_\delta$) included in $\Omega_\delta$, we have $R_\delta>R+\delta$

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