On the behavior of a capillary surface at a re-entrant corner

Changes in a domain's geometry can force striking changes in the capillary surface lying above it. Concus and Finn [1] first studied capillary surfaces above domains with corners, in the presence of gravity. Above a corner with interior angle θ satisfying θ < π — 2γ, they showed that a capillary surface making contact angle γ with the bounding wall must approach infinity as the vertex is approached. In contrast, they showed that for θ ̂ π — 2γ the solution u(xy y) is bounded, uniformly in θ as the corner is closed. Since their paper appeared, the continuity of u at the vertex has been an open problem in the bounded case. In this note we show by example that for any θ> 7Γ and any γ Φ π/2 there are domains for which u does not extend continuously to the vertex. This is in contrast to the case π> θ> π — 2γ; here independent results of Simon [5] show that u actually must extend to be C1 at the vertex. We consider bounded domains Ω in R2 with piecewise smooth boundaries dΩ, and functions u(x, y) satisfying ( i) div Tu = 2H(u) = icu in Ω; Tu = Du/]/l + Du \ H(u) = mean curvature of the surface z = u(x, y), K> 0. (ii) Tu-n = COST on the smooth part of 3Ω; 0 < £ 7 ^ π, n = ex-terior normal to dΩ. Physically u describes the capillary surface formed when a verti-cal cylinder with horizontal cross section Ω is placed in an infinite reservoir of liquid having rest height z = 0. Then σ where p = density of liquid g = (downward) acceleration of gravity σ — surface tension between liquid and air. cos 7 = —