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

Abstract

We investigate the manner in which a non-parametric surface $z=f(x,y)$ of prescribed mean curvature approaches its radial limits at a reentrant corner. We find, for example, that the solution $f(x,y)$ approaches a fixed value (an extreme value of its radial limits at the corner) as a Hölder continuous function with exponent $\frac{2}{3}$ as $(x,y)$ approaches the reentrant corner non-tangentially from inside a distinguished half-space. We also mention an application of our results to a problem in the production of capacitors involving "dip-coating."