The $$\infty $$-capacity and Faber–Krahn inequality on Grushin spaces

Abstract

In this paper we investigate the $$\infty $$ -capacity and the Faber–Krahn inequality on the Grushin space $${\mathbb {G}}^n_\alpha $$ . On the one hand we show that the $$\infty $$ -capacity equals the limit of the p-th root of the p-capacity as $$p\rightarrow \infty $$ and give a simple geometric characterization in terms of the Carnot–Caratheodory distance, on the other hand we establish some basic properties for the $$\infty $$ -capacity, $$\infty $$ -eigenvalue and the Faber–Krahn inequality on the Grushin space $${\mathbb {G}}^n_\alpha $$ .