Vol. 88, No. 2, 1980

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

Nicholas Jacob Korevaar

Vol. 88 (1980), No. 2, 379–385

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(x,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 > π and any γπ∕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.

Mathematical Subject Classification 2000
Primary: 49F10, 49F10
Secondary: 53A10, 76D15
Received: 16 November 1979
Published: 1 June 1980
Nicholas Jacob Korevaar