Boundary Behavior of a Nonparametric Surface of Prescribed Mean Curvature Near a Reentrant Corner

Abstract

ABSTRACT. Let 0 be an open set in R2 which is locally convex at each point of its boundary except one, say (0,0). Under certain mild assumptions, the solution of a prescribed mean curvature equation on Q behaves as follows: All radial limits of the solution from directions in 12 exist at (0,0), these limits are not identical, and the limits from a certain half-space (H) are identical. In particular, the restriction of the solution to Q n H is the solution of an appropriate Dirichlet problem. Introduction.We consider here the behavior of a generalized solution of the equation for surfaces of prescribed mean curvature at an inner corner of the boundary where the solution is discontinuous. This work is a generalization of the previous work of the second author [8], which dealt with the minimal surface equation. It was shown there that all radial limits exist and that they are constant in directions coming from a half-space. Here we find that the same result holds for (nonparametric) surfaces of prescribed mean curvature.

Behavior of a Bounded Non-Parametric $H$-Surface Near a Reentrant Corner

“…case (ii) of Proposition We may also determine the behavior of RI(S) as S approaches S i from below or 9 + ir from above. In fact: [2]) as indicated in the comments below preceeding the proof of Theorem 1 and an appropriate conformal map of the unit disk onto B, we find that there exists X E C°(B : R3 ) fl C2 (B : R3),…”

Radial limits of bounded nonparametric prescribed mean curvature surfaces

“…then we may parametrize the graph of f over Ω 1 in isothermal coordinates as above and the arguments in [2,6,9] can be used to show that c is uniformly continuous on Ω 1 and so extends to be continuous on Ω 1 (i.e. Let k : E \ B 1 ρ1(δ) ∪ B 2 ρ2(δ) → E be a conformal map.…”

“…We shall prove Theorem 1. Let f ∈ C 2 (Ω) satisfy (1) and suppose (2) holds and α ∈ π 2 , π . Then for each θ ∈ (−α, α), Rf (θ) def = lim r↓0 f (r cos(θ), r sin(θ)) exists and Rf (·) is a continuous function on (−α, α) which behaves in one of the following ways: (i) Rf : (−α, α) → IR is a constant function (i.e.…”

“…(1). Suppose (2) holds and m = lim ∂ − Ω * ∋x→O f (x) exists. Then for each θ ∈ (−α, α), Rf (θ) exists and Rf (·) is a continuous function on…”

On the relationship of continuity and boundary regularity in prescribed mean curvature Dirichlet problems

“…recall then that m > j a a+p 2 − b . Let f be the variational solution of (1)-(2) with Ω and φ as given here; that is, let f minimize the functional given in (3) and notice that the existence of f follows from (10), (16), §1.D. of [7] and [6,8].…”