Stability of Minimising Harmonic Maps under W^1,p Perturbations of Boundary Data: p≥ 2

2018/10/24 by Siran Li, Li, Siran
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1810.10599

Abstract

Let Ω⊂ ℝ³ be a Lipschitz domain, and consider a harmonic map v: Ω→ \mathbbS² 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 Hölder 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 \citeal and Mazowiecka--Strzelecki \citems.

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