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).

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[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)


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