Nonexistence and instability in the exterior Dirichlet problem for the minimal surface equation in the plane

Abstract

In this paper we investigate the Dirichlet problem \Ω is bounded. We sharpen previous non-existence results for this exterior Dirichlet problem by showing that even the smallness of the α-Holder norm, 0 < α < \ is not enough for the classical solvability of (1) and (2), not imposing any asymptotical conditions at infinity upon possible solutions. In particular, we explicitely exhibit smooth data / of arbitrary small C α -norm for which (1), (2) is not solvable in the space C°(Ω)ΠC 2 (Ω). The key idea of our proof is to replace the original problem (1), (2) on a known domain but with unknown boundary conditions at infinity by the corresponding problem on some unknown (bounded) domain, but with fixed boundary data. By the same method we show the instability of the exterior Dirichlet problem with respect to C a -small perturbations of the boundary data, 0 <a < ^ provided that Ω is the complement of a strictly convex set.

On solutions to the exterior Dirichlet problem for the minimal surface equation with catenoidal ends

“…@ / with Lipschitz constant K 2 OE0; 1= p n 1/ the Dirichlet problem is solvable if the oscillation of ' is small enough depending on n, K and and the constant 1= p n 1 is sharp, the borderline case K D 1= p n 1 apparently being open. An extension of this result in two dimensions was given in [12]. As far as we could see the previous authors did not bother about giving their smallness condition an explicit form.…”

“…We find it therefore useful to formulate (for n D 2/ Williams' condition in a more concise form which is suitable for a direct numerical evaluation, see Theorem 2.2 below. We use the same barriers as in [12], but in a more refined way.…”

“…It is shown in [20] and [12] that the bound 1 for the Lipschitz constant is optimal, the borderline case K D 1 apparently being open.…”

“…/ and uj @ D ' it suffices to construct barriers for the non-convex points, i.e., those points p 2 @ which possess no neighborhood U such that U \ is convex. We use the same kind of barriers as [12] but, in contrast to this paper, the present calculations will however lead to explicit conditions. According to our assumption, for such p 2 @ there is a disc D R .q/ D ¹z 2 R 2 j jz qj < Rº contained in R 2 n such that p 2 @D R .q/.…”

Untitled

“…To mention a few examples, compact convex hypersurfaces are considered in the closed case in [2], for Neumann boundary conditions in [7], and for Dirichlet boundary conditions in [4,5,8,10], based on the method of [12]. In the non-compact case, complete convex hypersurfaces are found in [1,9], whereas [6,13] deal with equations of mean curvature type in exterior domains. In this paper, we consider the Dirichlet problem for convex hypersurfaces of prescribed Gauß curvature in exterior domains.…”