Regularity at the free boundary for approximate H-surfaces in Riemannian manifolds (Q2135744)

The geometric set-up of this article is made up of a compact Riemannian surface \((M,g)\) with smooth boundary \(\partial M\), a compact \(d\)-dimensional Riemannian manifold \((N,h)\) and a \(k\)-dimensional (\(1\leq k\leq d-1\)) closed submanifold \(K\subset N\), to define the set \[ W^{1,2} (M,N;K) = \{ u\in W^{1,2} (M,N) : u(\partial M) \subset K \, \mbox{a.e.}\}. \] Then, given a \(C^1\) two-form \(\omega\) on \(N\) with \(|d\omega|_{L^\infty}\) finite and \(\omega|_{TK\times TK} \equiv 0\), one considers the functional \[ E^{\omega}(u) = \frac12 \int_M |\nabla u|^2 + u*\omega \] and its critical points, among maps in \(W^{1,2} (M,N;K)\), are called weakly \(H\)-surfaces. More generally, given a \(TN\)-valued map \(F\) on \(M\), approximate weakly \(H\)-surfaces are (weak) solutions of the system \[ \begin{cases} \Delta u - d\omega(\nabla^\perp u, \nabla u) + A(u)(\nabla u, \nabla u) = F \\ \left( \frac{\partial u}{\partial n}\right)^\perp = 0 , \end{cases}\tag{1} \] \(n\) being the outward unit normal vector field to the boundary \(\partial M\). The main result of this paper is to give a new, more geometrical, proof that if \(F\in L^p\) then a solution \(u\) to the above system (1), with free boundary \(u(\partial M) \subset K\), must be in \(W^{2,p}\). The method relies first on the conformal invariance of the functional to rewrite the problem on a half disk \(D^+\), with \(\{y=0\}\) being the free boundary. Then, the map is extended to \(D^-\) by geodesic reflection (originally due to \textit{R. Gulliver} and \textit{J. Jost} [J. Reine Angew. Math. 381, 61--89 (1987; Zbl 0619.35117)]) and shown to satisfy a new equation over the whole disk \(D\). This equation can then be formulated in the form \[ d^* ( A du) = \langle \Omega , A du\rangle + f \] where \(\Omega\) is an \({\mathbb R}^n\)-valued exterior form. Results on critical points of conformally invariant functionals imply \(W^{2,p}\) norm, hence regularity at the free boundary, and \(W^{2,p}\) estimates up to the free boundary.