Solutions of the Cheeger problem via torsion functions

Abstract

The Cheeger problem for a bounded domain Ω ⊂ R N , N > 1 consists in minimizing the quotients | ∂ E | / | E | among all smooth subdomains E ⊂ Ω and the Cheeger constant h ( Ω ) is the minimum of these quotients. Let ϕ p ∈ C 1 , α ( Ω ¯ ) be the p-torsion function, that is, the solution of torsional creep problem − Δ p ϕ p = 1 in Ω, ϕ p = 0 on ∂ Ω, where Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplacian operator, p > 1 . The paper emphasizes the connection between these problems. We prove that lim p → 1 + ( ‖ ϕ p ‖ L ∞ ( Ω ) ) 1 − p = h ( Ω ) = lim p → 1 + ( ‖ ϕ p ‖ L 1 ( Ω ) ) 1 − p . Moreover, we deduce the relation lim p → 1 + ‖ ϕ p ‖ L 1 ( Ω ) ⩾ C N lim p → 1 + ‖ ϕ p ‖ L ∞ ( Ω ) where C N is a constant depending only of N and h ( Ω ) , explicitely given in the paper. An eigenfunction u ∈ BV ( Ω ) ∩ L ∞ ( Ω ) of the Dirichlet 1-Laplacian is obtained as the strong L 1 limit, as p → 1 + , of a subsequence of the family { ϕ p / ‖ ϕ p ‖ L 1 ( Ω ) } p > 1 . Almost all t-level sets E t of u are Cheeger sets and our estimates of u on the Cheeger set | E 0 | yield | B 1 | h ( B 1 ) N ⩽ | E 0 | h ( Ω ) N , where B 1 is the unit ball in R N . For Ω convex we obtain u = | E 0 | − 1 χ E 0 .