reference for Poincaré constant of $W_0^{1,1}(B)$

Question

I'm trying to figure the optimal Poincaré constant of the space $W_0^{1,1}(B)$, Sobolov space on the unit ball in $\mathbb{R}^n$. Most paper I found is either about $W_0^{1,2}$ or $W^{1,2}$ with $0$ mean.

Is there any reference for $W_0^{1,1}(B)$ or $W_0^{1,1}(\Omega)$ for a general convex compact domain $\Omega$? Thanks

Answer 1

This minimizer is equivalent to the first nonlinear eigenvalue of $\mathbf{1}$-Laplacian, which is equal to the Cheeger constant of the domain $B$. The reference is here by B.Kawohl and M.Novaga, which studies the relation between the first non-linear eigenvalues.

Kawohl, B; Novaga, M, The (p)-Laplace eigenvalue problem as (p\to 1) and Cheeger sets in a Finsler metric, J. Convex Anal. 15, No. 3, 623-634 (2008). ZBL1186.35115..