On the evolution of harmonic mappings of Riemannian surfaces (Q1077769)

Let (M,\(\gamma)\) be a Riemannian surface with metric tensor \(\gamma =(\gamma_{\alpha \beta})_{1\leq \alpha,\beta \leq 2}\) and (N,g) an n- manifold with metric tensor \(g=(g_{ij})_{1\leq i,j\leq n}\). For differentiable mappings \(u: M\to N\) an energy is defined \[ E(u)=\int_{M}e(u)dM,\quad e(u)=(1/2)\gamma^{\alpha \beta}(x) g_{ij}(u) (\partial /\partial x^{\alpha})u^ i (\partial /\partial x^{\beta \quad})u^ j \] and the well-known equation for harmonic maps (1) \(-\Delta_ Mu=\Gamma (u)(\nabla u,\nabla u)_ M\) where \(\Gamma^{\ell}_{ij}\) are Christoffel symbols of the metric g and \(\Delta_ M\) is the Laplace-Beltrami operator. The aim of the paper is twofold. First, for the evolution problem associated with (1), the existence of a unique global solution for finite initial energy \(E(u_ 0)<\infty\) is established. Second, a local Palais-Smale type compactness result for the energy functional E which permits a direct proof of the Sacks-Uhlenbeck results is presented.