Existence of doubly connected minimal graphs in singular boundary configurations
Gudrun Turowski · Asymptotic Analysis · 2000
In this paper we prove the existence of nonparametric minimal surfaces of annulus type spanning boundary configurations (Γ ,S) where S is a vertical cylinder above a simple closed polygon P (S) in the x, y-plane and the surrounding Jordan curve Γ is given as a generalized graph above its convex projection curve P (Γ ) (cf. Corollary 1). The behaviour of such minimal surfaces at the edges of the support surface is of special interest. The case of more general support surfaces S where P (S) merely is a closed piecewise smooth Jordan curve can be treated by approximation. The nonparametric existence problem is by no means trivial. For instance, concerning the pure Dirichlet problem for the nonparametric minimal surface equation on a nonconvex Jordan domain Ω, there are always (arbitrary small) boundary values on ∂Ω for which the problem is not solvable. This result of Jenkins and Serrin can be found in the literature, see, e.g., [21, §411]. We have assumed P (Γ ) to be convex, but we try to solve the nonparametric minimal surface equation on the doubly connected domain G bounded by P (Γ ) and P (S), i.e., on a domain with a hole. Moreover we have a mixed boundary condition. In [20] T. Nehring proved the existence of an embedded parametric minimal surface with a Jordan curve Γ lying above a circle as fixed boundary and with a free boundary on a smooth surface F which bounds a compact, strictly convex body. In contrast to that paper we do not have to impose any convexity condition on the cylinder surface S. Finally, the support surface S is singular because we allow S to have edges. In the forthcoming paper [27] it will be shown that the free trace of a minimal surface spanning (Γ ,S) need not have a 1–1-projection into the x, y-plane. The free trace can attach to an edge of S in a full interval. This phenomenon is called edge-creeping. Apparently this phenomenon was discovered by Y.W. Chen when treating an exterior free boundary value problem for minimal surfaces (cf. [4]). There the support surface is a vertical cylinder above a closed convex polygon. Later S. Hildebrandt and J.C.C. Nitsche in [8] proved existence and optimal regularity of solutions to a partially free boundary value problem where edge-creeping occurs. A semifree problem for disc-type minimal surfaces with singular boundaries has been discussed in joint work of S. Hildebrandt and F. Sauvigny [12–14]. They choose the support surface S as the boundary of a wedge, or, more generally, as a polyhedral cylinder surface, and Γ is assumed to be a smooth Jordan arc with its endpoints on S. In [12] asymptotic expansion formulas for the derivatives Xw and Nw of the minimal surface X and its Gauss map N are proved, which can easily be transferred to the doubly connected case. In [14] an existence theorem for embedded minimal surfaces of disc type is given. This is achieved by combining the results of [11], obtained in the case of smooth boundaries, and [12] with a suitable