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- W2945752022 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 begin{equation}label{e:main0} begin{cases} u_t = Delta utext{ in }mathbb{R}^2_+times (0, T), u(x,0,t) in mathbb{S}^1text{ for all }(x,0,t)in partialmathbb{R}^2_+times (0, T), frac{du}{dy}(x,0,t)perp T_{u(x,0,t)}mathbb{S}^1text{ for all }(x,0,t)in partialmathbb{R}^2_+times (0, T), u(cdot, 0) = u_0text{ in }mathbb{R}^2_+ end{cases} end{equation} for a function $u:mathbb{R}^2_+times [0, T)to mathbb{R}^2$. Here $u_0 :mathbb{R}^2_+to mathbb{R}^2$ is a given smooth map and $perp$ stands for orthogonality. We prove the existence of initial data $u_0$ such that (ref{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}." @default.
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- W2945752022 date "2023-08-01" @default.
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- W2945752022 title "Singularity formation in the harmonic map flow with free boundary" @default.
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- W2945752022 doi "https://doi.org/10.1353/ajm.2023.a902958" @default.
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