On the free boundary of surfaces with bounded mean curvature: the non-perpendicular case (Q1275230)

The author considers the boundary behavior of a mapping \(X:B_{1}(0)\subset{\mathbb R}^{2}\to{\mathbb R}^{3},\) \(X\in H^{1,2},\) that is a minimizer or stationary point among mappings with partially free or free boundary values on a supporting surface \(S\) for the functional \[ \frac 12\iint_{B_{1}(0)}| \nabla X| ^{2} du dv + \iint_{B_{1}(0)}Q(X)\cdot(X_{u}\wedge X_{v}) du dv. \] Here \(Q\in C^{1}({\mathbb R}^{3},{\mathbb R}^{3})\) is a given vector field. A smooth solution is known to parametrize a surface of mean curvature \(\text{div }Q/2\) and to satisfy the free boundary condition \(| Q\cdot N| = \cos(\alpha),\) where \(\alpha\) denotes the angle in which \(X\) meets the supporting surface \(S\). The main result of this paper, Theorem 4.1, is that if a local smallness condition is imposed near \(S,\) then for all \(d\in(0,1)\) the minimizer \(X\) is a Hölder continuous function on \({\overline Z}_{d}:=\{w\in B^{1}(0):| w| <1-d\}\). This improves on a result of \textit{S. Hildebrandt} [Math. Ann. 194, 316-331 (1971; Zbl 0219.49015)]. To follow the arguments of this paper, the reader should be familiar with the appropriate parts of the treatise on minimal surfaces by \textit{U. Dierkes, S. Hildebrandt, A. Küster} and \textit{O. Wohlrab} [``Minimal surfaces I. Boundary value problems'' (1992; Zbl 0777.53012); ``Minimal surfaces II. Boundary regularity'' (1992; Zbl 0777.53013)] and with the paper of \textit{M. Grüter, S. Hildebrandt} and \textit{J. C. C. Nitsche} [Acta Math. 156, 119-152 (1986; Zbl 0609.49027)].