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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a Lipschitz domain, and consider a harmonic map $v:\mathrm{\Omega }\to {\mathbb{S}}^2$ with boundary data $v\mathrm{\mid }\mathrm{\partial }\mathrm{\Omega }=\phi$ which minimises the Dirichlet energy. For $p\ge 2$, we show that any energy minimiser $u$ whose boundary map $\psi$ has a small $W^{1,p}$-distance to $\phi$ is close to $v$ in H"{o}lder 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 \cite{al} and Mazowiecka–Strzelecki \cite{ms}.