Stability of Minimising Harmonic Maps under $W^{1,p}$ Perturbations of Boundary Data: $p\geq 2$
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| Format: | Preprint |
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2018
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| _version_ | 1866917284436508672 |
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| author | Li, Siran |
| author_facet | Li, Siran |
| contents | Let $Ω\subset \mathbb{R}^3$ be a Lipschitz domain, and consider a harmonic map $v: Ω\rightarrow \mathbb{S}^2$ with boundary data $v|\partialΩ= φ$ which minimises the Dirichlet energy. For $p\geq 2$, 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 \cite{al} and Mazowiecka--Strzelecki \cite{ms}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1810_10599 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Stability of Minimising Harmonic Maps under $W^{1,p}$ Perturbations of Boundary Data: $p\geq 2$ Li, Siran Differential Geometry 53C43, 58E20 Let $Ω\subset \mathbb{R}^3$ be a Lipschitz domain, and consider a harmonic map $v: Ω\rightarrow \mathbb{S}^2$ with boundary data $v|\partialΩ= φ$ which minimises the Dirichlet energy. For $p\geq 2$, 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 \cite{al} and Mazowiecka--Strzelecki \cite{ms}. |
| title | Stability of Minimising Harmonic Maps under $W^{1,p}$ Perturbations of Boundary Data: $p\geq 2$ |
| topic | Differential Geometry 53C43, 58E20 |
| url | https://arxiv.org/abs/1810.10599 |