Harmonic map heat flow with rough boundary data
Lu Wang · Transactions of the American Mathematical Society · 2012
Let B 1 B_1 be the unit open disk in R 2 \mathbb {R}^2 and M M a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in H 1 ( [ 0 , T ] × B 1 , M ) H^1([0,T]\times B_1,M) whose energy is non-increasing in time, given initial data u 0 ∈ H 1 ( B 1 , M ) u_0\in H^1(B_1,M) and boundary data γ = u 0 | ∂ B 1 \gamma =u_0|_{\partial B_1} . Previously, this uniqueness result was obtained by Rivière (when M M is the round sphere and the energy of initial data is small) and Freire (when M M is an arbitrary closed Riemannian manifold), given that u 0 ∈ H 1 ( B 1 , M ) u_0\in H^1(B_1,M) and