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

Abstract

Let $\Omega \subset {\mathbb{R}}^3$ be a Lipschitz domain, and consider a harmonic map $v:\Omega \to {\mathbb{S}}^2$ with boundary data $v|\partial \Omega =\phi$ 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 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 \cite{al} and Mazowiecka--Strzelecki \cite{ms}.

arXiv commentV3 contains more elaboration in Section 3, p.7 on the usage of L. Simon's asymptotic theory than the published version (to appear in JDE, 2021)

Cited by in corpus (1)

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