Improved regularity of harmonic map flows with H�lder continuous energy

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 Holder continuous, then the weak limit u(T) must also be Holder continuous.