Minimal extension for the α-Manhattan norm

Abstract

Let ∂Q be the boundary of a convex polygon in R2, eα=(cosα,sinα) and eα⊥=(−sinα,cosα) a basis of R2 for some α∈[0,2π) and φ:∂Q→R2 a continuous, finitely piecewise linear injective map. We construct a finitely piecewise affine homeomorphism v:Q→R2 coinciding with φ on ∂Q such that the following property holds: ∣⟨Dv,eα⟩∣(Q) (resp., ⟨Dv,eα⊥⟩∣(Q)) is as close as we want to inf∣⟨Du,eα⟩∣(Q) (resp., inf∣⟨Du,eα⊥⟩∣(Q)) where the infimum is meant over the class of all BV homeomorphisms u extending φ inside Q. This result extends that already proven by Pratelli and the third author in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 3, 511–555] in the shape of the domain.