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Weak convergence of wave maps from \((1+2)\)-dimensional Minkowski space to Riemannian manifolds. (English) Zbl 0906.35061
A wave map is akin to a harmonic map and is defined here as a mapping \(u\) of the \((1+2)\)-dimensional Minkowski spacetime \(\mathbb{R}\times M\) into a closed Riemannian manifold \(N\), which is isometrically embedded in a Euclidean space \(\mathbb{R}^d\), such that \(\square u\) is orthogonal to the tangent space \(T_uN\) in \(\mathbb{R}^d\). Equivalently, \(\square u\) equals some nonlinear expression in \(u\). Moreover, some energy estimate admits a weak, i.e. distributional, formulation in terms of differential forms.
It is shown that the weak limit of a sequence of wave maps with uniformly bounded energy is again a wave map. Ingredients of the proof are a reduction to periodic fields \(u\), a Coulomb gauge, a concise description of the concentration set of a sequence of wave maps, and the use of \({\mathcal H}^1\)-BMO duality.

MSC:
35L70 Second-order nonlinear hyperbolic equations
58E20 Harmonic maps, etc.
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