Improved regularity of harmonic map flows with Hölder continuous energy
Improved regularity of harmonic map flows with Hölder continuous energy
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利用 Hölder 连续能量改进谐波映射流的规律性
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发表时间：
2004
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通讯作者：
P. Topping
中科院分区：
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作者：
P. Topping
摘要：对于平滑调和映射流 $u: mathcal{M} imes [0,T) omathcal{N}$ 且膨胀为 $tuparrow T$，有人询问[5,6,7]弱极限 $u(T): mathcal{M} omathcal{N}$ 是否连续。最近，在[12]中，我们表明通常不需要如此。同时，能量函数$E(u(cdot)): [0,T) o mathbb{R}$具有弱正、平滑、弱递减的特性，连续扩展到[0,T]。这里我们证明，如果这个扩展也是 Hölder 连续的，那么弱极限 u(T) 也必须是 Hölder 连续的。
Abstract.For a smooth harmonic map flow $u: mathcal{M} imes [0,T) omathcal{N}$ with blow-up as $tuparrow T$, it has been asked [5,6,7] whether the weak limit $u(T): mathcal{M} omathcal{N}$ is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function $E(u(cdot)): [0,T) o mathbb{R}$, 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.