Boundary regularity of weakly anchored harmonic maps

arXiv (Cornell University), 2018

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 K ⊂ N satisfying sup n ∇u n L 2 (M) + τ(u n) L 2 (M) ≤ Λ, where τ(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. 1. introduction Let (M, g) be a compact Riemannian manifold with smooth boundary and (N, h) be a compact Riemannian manifold of dimension n. Let K ⊂ N be a k−dimensional closed submanifold where 1 ≤ k ≤ n. For a mapping u ∈ C 2 (M, N), the energy density of u is defined by e(u) = 1 2 |∇u| 2 = Trace g u * h, where u * h is the pull-back of the metric tensor h. The energy of the mapping u is defined as E(u) = M e(u)dvol g. Define C(K) = u ∈ C 2 (M, N); u(∂M) ⊂ K. A critical point of the energy E over C(K) is a harmonic map with free boundary u(∂M) on K. The problem of the existence, uniqueness and regularity of such harmonic maps with a free boundary was first systematically investigated in [8]. By Nash's embedding theorem, (N, h) can be isometrically embedded into some Euclidean space R N. Then we can get the Euler-Lagrange equation ∆ g u = A(u)(∇u, ∇u),

Archive for Rational Mechanics and Analysis, 2014

We study tensor-valued minimizers of the Landau-de Gennes energy functional on a simply-connected planar domain Ω with noncontractible boundary data. Here the tensorial field represents the second moment of a local orientational distribution of rod-like molecules of a nematic liquid crystal. Under the assumption that the energy depends on a single parameter-a dimensionless elastic constant ε > 0we establish that, as ε → 0, the minimizers converge to a projectionvalued map that minimizes the Dirichlet integral away from a single point in Ω. We also provide a description of the limiting map.

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

We consider the problem of minimizing the energy functional R (|∇u| 2 + χ {u>0} ). We show that the singular axisymmetric critical point of the functional is an energy minimizer in dimension 7. This is the first example of a non-smooth energy minimizer. It is analogous to the Simons cone, a least area hypersurface in dimension 8.

Rendiconti Lincei - Matematica e Applicazioni, 2000

Indiana University Mathematics Journal, 2018

The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the geometric-driven profile of ground states. In this work we study, under very general assumptions, the convergence of minimizers towards harmonic maps. We show that the convergence is locally uniform up to the boundary, away from the lower dimensional singular set. Our results generalize related findings, most notably in the theory of liquid-crystals, to all dimensions n ≥ 3, and to general nonlinearities. Our proof follows a well-known scheme, relying on small energy estimate and monotonicity formula. It departs substantially from previous studies in the treatment of the small energy estimate at the boundary, since we do not rely on the specific form of the potential. In particular this extends existing results in 3-dimensional settings. In higher dimensions we also deal with additional difficulties concerning the boundary monotonicity formula.

Analysis & PDE

We study minimizers of a family of functionals E ε indexed by a characteristic length scale ε, whose formal limit is E ⋆ (u) =´W (∇u) for u taking values into a manifold, where W is a positive definite quadratic form: minimizers of E ⋆ are W-harmonic mapsclassical harmonic maps correspond to the isotropic case W (∇u) = |∇u| 2. We show that the convergence of minimizers of E ε to W-harmonic maps is locally uniform outside a singular set. We treat general energies, covering in particular the 3D Landau-de Gennes model for liquid crystals, with three distinct elastic constants. In the isotropic case, similar results are standard and rely on three ingredients: a monotonicity formula for the scale-invariant energy on small balls, a uniform pointwise bound, and a Bochner equation for the energy density. In the level of generality we consider in this work, all of these ingredients are absent and we have to use a very different strategy. Finding ways around the lack of monotonicity formula is particularly interesting, since this issue is why optimal estimates on the singular set of W-harmonic maps constitute an open problem. Our novel argument relies on showing appropriate decay for the energy on small balls, separately at scales smaller and larger than ε: the former is obtained from the regularity of solutions to elliptic systems while the latter is inherited from the regularity of W-harmonic maps. This also allows us to handle physically relevant boundary conditions for which, even in the isotropic case, uniform convergence up to the boundary was open.

2019

We study the asymptotic behavior, when $\varepsilon\to0$, of the minimizers $\{u_\varepsilon\}_{\varepsilon>0}$ for the energy \begin{equation*} E_\varepsilon(u)=\int_{\Omega}\Big(|\nabla u|^2+\big(\frac{1}{\varepsilon^2}-1\big)|\nabla|u||^2\Big), \end{equation*} over the class of maps $u\in H^1(\Omega,{\mathbb R}^2)$ satisfying the boundary condition $u=g$ on $\partial\Omega$, where $\Omega$ is a smooth, bounded and simply connected domain in ${\mathbb R}^2$ and $g:\partial\Omega\to S^1$ is a smooth boundary data of degree $D\ge1$. The motivation comes from a simplified version of the Ericksen model for nematic liquid crystals with variable degree of orientation. We prove convergence (up to a subsequence) of $\{u_\varepsilon\}$ towards a singular $S^1$-valued harmonic map $u_*$, a result that resembles the one obtained in \cite{BBH} for an analogous problem for the Ginzburg-Landau energy. There are however two striking differences between our result and the one involving the Gin...

Archive for Rational Mechanics and Analysis, 2010

We prove that, if u : Ω → R N is a solution to the Dirichlet variational problem

Mathematische Zeitschrift, 1976

Mathematische Zeitschrift, 1993