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Article overview
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Singularity formation in the harmonic map flow with free boundary | | Yannick Sire
; Juncheng Wei
; Youquan Zheng
; | | Date: |
15 May 2019 | | Abstract: | In the past years, there has been a new light shed on the harmonic map
problem with free boundary in view of its connection with nonlocal equations.
Here we fully exploit this link, considering the harmonic map flow with free
boundary egin{equation}label{e:main0} egin{cases} u_t = Delta u ext{ in
}mathbb{R}^2_+ imes (0, T),\ u(x,0,t) in mathbb{S}^1 ext{ for all
}(x,0,t)in partialmathbb{R}^2_+ imes (0, T),\ frac{du}{dy}(x,0,t)perp
T_{u(x,0,t)}mathbb{S}^1 ext{ for all }(x,0,t)in partialmathbb{R}^2_+ imes
(0, T),\ u(cdot, 0) = u_0 ext{ in }mathbb{R}^2_+
end{cases} end{equation} for a function $u:mathbb{R}^2_+ imes [0, T) o
mathbb{R}^2$. Here $u_0 :mathbb{R}^2_+ o mathbb{R}^2$ is a given smooth map
and $perp$ stands for orthogonality. We prove the existence of initial data
$u_0$ such that (
ef{e:main0}) blows up at finite time with a profile being
the half-harmonic map. This answers a question raised by Yunmei Chen and
Fanghua Lin in Remark 4.9 of cite{ChenLinJGA1998}. | | Source: | arXiv, 1905.5937 | | Services: | Forum | Review | PDF | Favorites | |
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