Stability of minimising harmonic maps under W1, perturbations of boundary data: p ≥ 2

Let Ω ⊂ R 3 be a Lipschitz domain. Consider a harmonic map v : Ω → S 2 with boundary data v | ∂ Ω = φ which minimises the Dirichlet energy. For p ≥ 2 , we show that any energy minimiser u whose boundary map ψ has a small W 1 , p -distance to φ is close to v in Holder norm modulo bi-Lipschitz homeomorphisms, provided that v is the unique minimiser attaining the boundary data. The index p = 2 is sharp: the above stability result fails for p 2 due to the constructions by Almgren–Lieb [2] and Mazowiecka–Strzelecki [15] .