## Relaxed energies for harmonic maps.(English)Zbl 0793.58011

Variational methods, Proc. Conf., Paris/Fr. 1988, Prog. Nonlinear Differ. Equ. Appl. 4, 37-52 (1990).
[For the entire collection see Zbl 0713.00009.]
Let $$\Omega \subset \mathbb{R}^ 3$$ be an open bounded set with smooth boundary $$\partial \Omega$$. Set $H^ 1(\Omega;S^ 2) = \{u \in H^ 1(\Omega;\mathbb{R}^ 3): | u(x)| = 1 \text{a.e.\} and}$
$H^ 1_ \varphi(\Omega;S^ 2) = \{u\in H^ 1(\Omega;S^ 2): u = \varphi \text{ on }\partial \Omega\},$ where $$\varphi: \partial \Omega \to S^ 2$$ is a given boundary datum.
The aim of this article is to establish some basic properties of the functionals $$L(u)$$ and $$F(u)$$ defined below. The interest of this approach lies on its applications to the theory of liquid crystals and to the study of singularities of harmonic maps into spheres. $L(u) = {1\over 4\pi}\displaystyle{\sup_{\substack{ \xi: \Omega\to \mathbb{R}\\ \| \nabla\xi \|_ \infty \leq 1}} }\left \{\int_ \Omega D(u)\cdot \nabla \xi - \int_{\partial \Omega} D(u) \cdot \nu \xi d\sigma\right\},$ where $$\nu$$ denotes the outward normal to $$\partial \Omega$$ and the vector field $$D(u)$$ is given by $$D(u) = (u.u_ y \wedge u_ z,\;u.u_ z \wedge u_ x,\;u.u_ x \wedge u_ y)$$.
$$F(u) = E(u) + 8 \pi L(u)$$, where $$E(u)$$ is the usual energy for harmonic maps.
By way of example, let us quote one result obtained in this paper:
Theorem: $$F$$ is sequentially lower semi-continuous on $$H^ 1_ \varphi$$ for the weak $$H^ 1$$ topology.
Reviewer: A.Ratto (Brest)

### MSC:

 58E20 Harmonic maps, etc. 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

regularity; liquid crystals; singularities; harmonic maps; spheres

### Citations:

Zbl 0793.58012; Zbl 0713.00009