Evolution of harmonic maps with Dirichlet boundary conditions
The global existence of weak flows of harmonic maps between two compact Riemannian manifolds with boundary \(M\) and \(N\) with Dirichlet boundary conditions is proved. This weak solution is smooth off a closed subset in \(\overline M\times \mathbb{R}_+\), which has a locally finite \(m\)-dimensional Hausdorff measure with respect to the parabolic metric. Moreover, as \(t\to +\infty\) suitably, the solution \(u(t)\) converges weakly in \(H^1(M, N)\) to a harmonic map \(u_\infty: M\to N\) with \(u_\infty|_{\partial M}= u_0|_{\partial M}\), which is smooth off a set in \(\overline M\), whose \((m- 2)\)-dimensional Hausdorff measure can be bounded in terms of the \(C^2\)-norm of \(u_0\) and the energy \(E(u_0)\).
scientific article; zbMATH DE number 836423