Freire, Alexandre; Müller, Stefan; Struwe, Michael Weak convergence of wave maps from \((1+2)\)-dimensional Minkowski space to Riemannian manifolds. (English) Zbl 0906.35061 Invent. Math. 130, No. 3, 589-617 (1997). 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. Reviewer: R.Schimming (Greifswald) Cited in 10 Documents MSC: 35L70 Second-order nonlinear hyperbolic equations 58E20 Harmonic maps, etc. Keywords:\({\mathcal H}^1\)-BMO duality; energy estimate PDFBibTeX XMLCite \textit{A. Freire} et al., Invent. Math. 130, No. 3, 589--617 (1997; Zbl 0906.35061) Full Text: DOI