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Let $\Omega \subset \mathbb{R}^3$ be a Lipschitz domain, and consider aharmonic map $v: \Omega \rightarrow \mathbb{S}^2$ with boundary data$v|\partial\Omega = \varphi$ which minimises the Dirichlet energy. For $p\geq2$, we show that any energy minimiser $u$ whose boundary map $\psi$ has a small$W^{1,p}$-distance to $\varphi$ is close to $v$ in H\"{o}lder norm modulobi-Lipschitz homeomorphisms, provided that $v$ is the unique minimiserattaining the boundary data. The index $p=2$ is sharp: the above stabilityresult fails for $p<2$ due to the constructions by Almgren--Lieb \cite{al} andMazowiecka--Strzelecki \cite{ms}.

Stability of Minimising Harmonic Maps under $W^{1,p}$ Perturbations of Boundary Data: $p\geq 2$