On a singular behavior of capillary surfaces (Q1424343)

Let \(D_1\) be a square, \(D_0\) is an inscribed disk, and the intermediate domains \(D_t\) are obtained by smoothing the corners of \(D_1\) with circular arcs, so that \(D_t\) converges to \(D_0\) as \(t\to0\). \textit{R. Finn} and \textit{A. A. Kosmodem'yanskii jun.} [Pac. J. Math 205, No.~1, 119--137 (2002; Zbl 1059.53010)] have shown that the heights \(u^t\) of capillary surfaces in vertical tubes with the sections \(D_t\) in a gravity field \(g\) satisfy \(\lim_{g\to0}\{\inf_{D_t} u^1-\sup_{D_t} u^t\}=\infty\) for every \(t\in(0,1)\), if \(u^1<u^0\) over \(D_0\) for all \(g>0\). This surprising behaviour contradicts the so-called common sense which suggests that a capillary tube of cross-section \(\Omega\) raises liquid higher over its section than does a tube of cross-section which contains \(\Omega\). Here the most general convex \(D_1\) is characterized that leads to such a discontinuous transition when \(D_0\) is a disk. In all these examples the boundaries of the domains \(D_t\) have a discontinuity in their curvatures. It is shown that this discontinuity cannot be viewed as the root cause the surprising behaviour of capillary surfaces. More presicely, an example is given in which one can choose domains \(D_t\) concentric to the disk \(D_0\). In the example the domain \(D_1\) is star-shaped with respect to the center of \(D_0\). The proof of the described behaviour is mainly based on comparison properties of capillary surfaces due to Finn and Kosmodem'yanskii, jun. [loc. cit.].