The limit as $p\rightarrow \infty$ for the eigenvalue problem of the 1-homogeneous $p$-Laplacian

Abstract

In this paper we study asymptotics as $p\rightarrow \infty$ of the Dirichlet eigenvalue problem for the $1$-homogeneous $p$-Laplacian, that is,

$$\begin{aligned} \left\{ \begin{array}{ll} -\frac{1}{p} |D u|^{2-p}\mathrm{div}\,(|D u|^{p-2}Du)=\lambda u, &{}\text{ in }\;\Omega ,\\ u=0,&{}\text{ on }\;\partial \Omega . \end{array}\right. \end{aligned}$$

Here $\Omega$ is a bounded starshaped domain in $\mathbb{R }^n$ and $p>n$. There exists a principal eigenvalue $\lambda _{1,p} (\Omega )$, which is positive, and has associated a non-negative nontrivial eigenfunction. Moreover, we show that $\lim _{p\rightarrow \infty }\lambda _{1,p}(\Omega )= \lambda _{1,\infty }(\Omega )$, where $\lambda _{1,\infty }(\Omega )$ is the first eigenvalue corresponding to the $1$-homogeneous infinity Laplacian, that is, $-\left( D^2u\frac{Du}{|Du|}\right) \cdot \frac{Du}{|Du|} =\lambda u$.