Approximate Harmonic Maps Research Articles

$$L^p$$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions

We establish an optimal $$L^p$$ -regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions $$n\ge 5$$ : $$\begin{aligned} \Delta ^2 u=\Delta (D\cdot \nabla u)+\text {div}(E\cdot \nabla u) +(\Delta \Omega +G)\cdot \nabla u +f \qquad \ \mathrm{{in}}\ B^n, \end{aligned}$$ where $$\Omega \in W^{1,2}(B^n, so_m)$$ is antisymmetric and $$f\in L^p(B^n)$$ , and $$D, E, \Omega , G$$ satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of $$\nabla u$$ and $$\nabla ^2 u$$ . This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm–Rivière, Struwe, and Wang. In particular, our results improve Struwe’s Hölder regularity theorem to any Hölder exponent $$\alpha \in (0,1)$$ when $$f\equiv 0$$ , and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of our techniques, we also partially extend the $$L^p$$ -regularity theory of approximate harmonic maps by Moser to Rivière-Struwe’s second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions $$n\ge 2$$ when $$p>\frac{n}{2}$$ , which partially confirms an interesting expectation by Sharp.

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

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

The qualitative behavior at the free boundary for approximate harmonic maps from surfaces

Let {u_n} be a sequence of maps from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold N with free boundary on a smooth submanifold Ksubset N satisfying supn‖∇un‖L2(M)+‖τ(un)‖L2(M)≤Λ,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sup _n \\ \\left( \\Vert \\nabla u_n\\Vert _{L^2(M)}+\\Vert \\tau (u_n)\\Vert _{L^2(M)}\\right) \\le \\Lambda , \\end{aligned}$$\\end{document}where tau (u_n) is the tension field of the map u_n. We show that the energy identity and the no neck property hold during a blow-up process. The assumptions are such that this result also applies to the harmonic map heat flow with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time.

Stratification for the singular set of approximate harmonic maps

The aim of this note is to extend the results in arXiv:1504.02043 to the case of approximate harmonic maps. More precisely, we will proved that the singular strata $S^k(u)$ of an approximate harmonic map are k-rectifiable, and we will show effect bounds on the quantitative strata. In the process we will simplify many of the arguments from arXiv:1504.02043, and in particular we produce a new main covering lemmas which vastly simplifies the older argument.

Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

For a sequence of approximate harmonic maps (u_n,v_n) (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form g_N -beta dt^2 for some Riemannian metric g_N and some positive function beta on N, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.

Boundary Bubbling Analysis of Approximate Harmonic Maps Under Either Weak or Strong Anchoring Conditions in Dimension 2

Boundary Bubbling Analysis of Approximate Harmonic Maps Under Either Weak or Strong Anchoring Conditions in Dimension 2

Remarks on approximate harmonic maps in dimension two

For the class of approximate harmonic maps \(u\in W^{1,2}(\Sigma ,N)\) from a closed Riemmanian surface \((\Sigma ,g)\) to a compact Riemannian manifold (N, h), we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps \(\{u_n\}:\Sigma \rightarrow N\), with tension fields \(\tau (u_n)\) bounded in the Morrey space \(M^{1,\delta }(\Sigma )\) for some \(0\le \delta <2\); and (ii) if an approximate harmonic map u has tension field \(\tau (u)\in L\log L(\Sigma )\cap M^{1,\delta }(\Sigma )\) for some \(0\le \delta <2\), then \(u\in W^{2,1}(\Sigma , N)\). Based on these estimates, we further establish the bubble tree convergence, referring to energy identity both \(L^{2,1}\) of gradients and \(L^1\)-norm of hessians and the oscillation convergence, for a weakly convergent sequence of approximate harmonic maps \(\{u_n\}\), with tension fields \(\tau (u_n)\) uniformly bounded in \(M^{1,\delta }(\Sigma )\) for some \(0\le \delta <2\) and uniformly integrable in \(L\log L(\Sigma )\).

Energy identity for approximate harmonic maps from surfaces to general targets

Let un be a sequence of mappings from a closed Riemannian surface M to a general Riemannian manifold N. If un satisfiessupn⁡(‖∇un‖L2(M)+‖τ(un)‖Lp(M))≤Λfor some p>1, where τ(un) is the tension field of un, then the so called energy identity and neckless property hold during blowing up. This result is sharp by Parker's example, where the tension fields of the mappings from Riemannian surface are bounded in L1(M) but the energy identity fails.

The blow-up analysis on two dimensional harmonic maps

In this paper, we recall and summarize that one has made some important progress in the blow-up analysis on two dimensional harmonic maps. We introduce the ``Ding-Trick and show how to use it to study the convergence behaviors of the necks as a sequence of harmonic maps or approximate harmonic maps from a closed Riemann surface blows up. We also announce some new results on energy identity obtained recently, and pose some open problems on two dimensional harmonic maps.

Bubble tree for approximate harmonic maps

In this paper, we set up the complete bubble tree theory for approximate harmonic maps from a Riemann surface with tension fields bounded in Zygmund class L ln + ⁡ L L\ln ^+ L . Some special cases of this theory have previously been used in a number of papers. On the other hand, one can see that this bubble tree theory is not true for the general target manifold if we only assume that the tension fields are bounded in L 1 L^1 uniformly.

No neck for approximate harmonic maps to the sphere

In this paper, we consider a sequence of maps from a Riemann surface to a standard sphere with tension fields bounded in an Orlicz space ϕ(L) where ϕ(L)={f:∫ϕ(∣f∣)<∞}. If there holds limt→∞ϕ(t)tlnt=∞, we can prove that there is no neck during blow up. This result improves our previous theorems in Li and Zhu [1,2]. One can see that our result here is optimal in the size conditions.

Remarks on approximate harmonic maps

Remarks on approximate harmonic maps

Energy identity for a class of approximate harmonic maps from surfaces

Energy identity for a class of approximate harmonic maps from surfaces