The evolution of harmonic mappings with free boundaries (Q757923)

The author considers the free boundary value problem for harmonic maps between Riemannian manifolds. Let M be a compact Riemannian manifold with boundary \(\partial M\) and N a smooth submanifold on \({\mathbb{R}}^ n\), \(\Sigma\) another submanifold of \({\mathbb{R}}^ n\) and \(S=N\cap \Sigma\). A solution to the free boundary problem is a harmonic map u: \(M\to N\) such that u(\(\partial M)\subset S\) and (\(\partial u/\partial n)\perp TS\) along \(\partial M\), i.e. an extremal of the energy functional on the space of maps u: \(M\to N\) such that u(\(\partial M)\subset S.\) To proceed, the author has to introduce technical restrictions: he supposes that M is a unit ball, and that there is a ball \(U\subset {\mathbb{R}}^ n\), containing N, such that \(\partial U\) is orthogonal to \(\Sigma\) and the projection \(\pi_{\Sigma}: U\to \Sigma \cap U\) is smooth in U and satisfies \(| D^ 2\pi_{\Sigma}| \cdot diam U<1/2\). An example of that situation is \(N=S^{n-1}\subset {\mathbb{R}}^ n\), \(\Sigma ={\mathbb{R}}^ k\times \{0\}\) or a small perturbation of \({\mathbb{R}}^ k\times \{0\}\). Then, the associated heat equation possesses a weak solution for all time, smooth off a set of codimension \(\geq 2\). As time goes to infinity, the solution converges weakly to a weak harmonic map smooth off a set of codimension 2. The author conjectures that the technical hypothesis can be removed, but for dim \(M\geq 3\) the free boundary conditions pose very difficult problems.