Bubbling analysis near the Dirichlet boundary for approximate harmonic maps from surfaces

Abstract

For a sequence of maps with a Dirichlet boundary condition from a compact Riemann surface with smooth boundary to a general compact Riemannian manifold, with uniformly bounded energy and with uniformly L 2 -bounded tension field, we show that the energy identity and the no neck property hold during a blow-up process near the Dirichlet boundary. We apply these results to the two dimensional harmonic map flow with Dirichlet boundary and prove the energy identity at finite and infinite singular time. Also, the no neck property holds at infinite time.Then u is a harmonic map if and only if τ(u) = 0.

Energy identity for a class of approximate Dirac-harmonic maps from surfaces with boundary

“…d n = dist(x n , ∂ 0 D + ) = |x n − x ′ n |. Similar to the boundary blow-up cases for approximate harmonic maps studied in [15,16], we…”

“…Firstly, we prove a small energy regularity theorem for the boundary case. For similar results for approximate harmonic maps, one can refer to the main estimate 3.2 in [27] and Lemma 2.1 in [8] for the interior case and one can also refer to Lemma 4.1 in [15], Lemma 2.4 in [16] for various boundary cases. 4 3 < q ≤ 2, and with boundary data (1.9), satisfying…”

“…Next we shall derive a Pohozaev type identity for approximate Dirac-harmonic maps with boundary data, extending the interior case given in Lemma 2.3 in [17]. For corresponding results for two dimensional approximate harmonic maps, one can refer to Lemma 2.4 [21] for the interior case and refer to Lemma 4.3 in [15] and Lemma 2.5 in [16] for various boundary cases.…”

“…Therefore, we just need to estimate the energy concentration in Ω 2 . Here, we use a similar method as in [15,16].…”

“…For approximate harmonic maps in dimension two, one can refer to e.g. [27,11,24,25,8,26,21,22,19,30,29] for the interior blow-up case and [15,16,10] for the boundary blow-up cases under various boundary constraints.…”