Uniqueness for the harmonic map flow in two dimensions

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Abstract

LetM be a two-dimensional Riemannian manifold with smooth (possibly empty) boundary. Ifu andv are weak solutions of the harmonic map flow inH 1(M×[0,T]; SN) whose energy is non-increasing in time and having the same initial data u0 ε H1(M,SN) (and same boundary values γ εH 3/2(∂M; SN) if ∂M; SN ≠Ø) thenu=v.