Abstract.
For a smooth harmonic map flow \(u: \mathcal{M}\times [0,T)\to\mathcal{N}\) with blow-up as \(t\uparrow T\), it has been asked [5,6,7] whether the weak limit \(u(T): \mathcal{M}\to\mathcal{N}\) is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function \(E(u(\cdot)): [0,T)\to \mathbb{R}\), being weakly positive, smooth and weakly decreasing, has a continuous extension to [0,T]. Here we show that if this extension is also Hölder continuous, then the weak limit u(T) must also be Hölder continuous.
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Received: 1 September 2003, Accepted: 7 October 2003, Published online: 25 February 2004
Version of 19/9/03. Partly supported by an EPSRC Advanced Research Fellowship.
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Topping, P. Improved regularity of harmonic map flows with Hölder continuous energy. Cal Var 21, 47–55 (2004). https://doi.org/10.1007/s00526-003-0246-5
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DOI: https://doi.org/10.1007/s00526-003-0246-5