Boundary behavior of capillary surfaces possibly with extremal boundary angles

Fei-Tsen Liang

doi:10.1155/ijmms.2005.3925

Abstract

For solutions to the capillarity problem possibly with the boundary contact angleθbeing0and/orπin a relatively open portion of the boundary which isC2, we will show that if the solution is locally bounded up to this portion of boundary, the trace of the solution on this portion is piecewise Lipschitz continuous and the solution is Hölder continuous up to the boundary, provided the prescribed mean curvature is bounded from above and from below. In the case whereθis not required to be bounded away fromπ/2,0, andπ, and the mean curvatureH(x,t0)belongs toLp(Ω)for somet0∈ℝandp>n, under the assumption that in a neighborhood of a relatively open portion of the boundary the solution is of rotational symmetry, the trace of the solution on this portion of the boundary is shown to be Hölder continuous with exponent1/nifn≥3and with exponent1/3ifn=2.

Highlights

Given a domain Ω ⊂ Rn, we are interested in regularity near the boundary ∂Ω for solutions u ∈ C2(Ω) of the mean curvature equation div Tu = H(x,u) on Ω, (1.1)
We are looking for a function u on Ω whose graph has the prescribed mean curvature H and which meets the cylinder over the boundary in the prescribed angle θ
We will show in this paper that if this portion of ∂Ω is C2 and if the prescribed mean curvature is bounded from above and from below, if the solution is bounded locally up to this portion of the boundary, the trace of the solution on this portion is piecewise Lipschitz continuous and the solution is Holder continuous up to the boundary

Summary

Introduction

We will show in this paper that if this portion of ∂Ω is C2 and if the prescribed mean curvature is bounded from above and from below, if the solution is bounded locally up to this portion of the boundary, the trace of the solution on this portion is piecewise Lipschitz continuous and the solution is Holder continuous up to the boundary. Suppose θ ≡ 0 or θ ≡ π in a relatively open subset ∂Ω of ∂Ω and Ω ⊂ Rn. If ∂Ω is C2 and if the prescribed mean curvature H(x, t) is locally Lipschitz continuous on Ω × R, satisfies (1.4), and is bounded in absolute value by a constant H∗, H(x, u) ≤ H∗, for x ∈ Ω. Suppose that in a neighborhood U of ∂Ω in Ω, the function u|U is of rotational symmetry, and suppose that the prescribed mean curvature H(x,t) is locally Lipschitz continuous on Ω × R, satisfies (1.4), and. The Holder norm is determined by n and (supΩ u − infΩ u), L, and H(x0)

Preliminaries

The Lipschitz continuity of the trace on the boundary