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Singular perturbation of manifold-valued maps with anisotropic energy

### Andres Contreras and Xavier Lamy

Vol. 15 (2022), No. 6, 1531–1560
##### Abstract

We establish small energy Hölder bounds, uniform with respect to $𝜀\in \left(0,1\right)$, for minimizers ${u}_{𝜀}$ of

 ${E}_{𝜀}\left(u\right):={\int }_{\mathrm{\Omega }}W\left(\nabla u\right)+\frac{1}{{𝜀}^{2}}{\int }_{\mathrm{\Omega }}f\left(u\right),$

where $W$ is a positive definite quadratic form and the potential $f$ constrains $u$ to be close to a given manifold $\mathsc{𝒩}$. This implies that, up to subsequence, ${u}_{𝜀}$ converges locally uniformly to an $\mathsc{𝒩}$-valued $W$-harmonic map, away from its singular set. This is the first result of its kind for general anisotropic energies covering in particular the previously open case of three-dimensional Landau–de Gennes model for liquid crystals, with three distinct elastic constants. Similar results in the isotropic case $W\left(\nabla u\right)=|\nabla u{|}^{2}$ 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; all of these ingredients are absent for general anisotropic $W$’s. In particular, the lack of monotonicity formula is an important reason why optimal estimates on the singular set of $W$-harmonic maps constitute an open problem. To circumvent these difficulties we devise an argument that 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.

##### Keywords
singular perturbation, liquid crystals
##### Mathematical Subject Classification 2010
Primary: 35J47, 35J50