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

Pablo Blanc

doi:10.3934/cpaa.2020158

Article Contents

2020, Volume 19, Issue 7: 3613-3623. Doi: 10.3934/cpaa.2020158

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

Pablo Blanc

Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina.

Received: May 31, 2019

Revised: December 31, 2019

Published: April 2020

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

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Abstract

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.

Keywords:

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

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

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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

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

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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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