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, \( -\left( D^2u\frac{Du}{|Du|}\right) \cdot \frac{Du}{|Du|} =\lambda u\).