A new example of singular height transition for capillary surfaces

Robert Finn
December 2003
Calculus of Variations & Partial Differential Equations;Dec2003, Vol. 19 Issue 1, p107
Academic Journal
Finn and Kosmodem?yanskii, Jr. gave an example of a domain ${\cal D}_{1}$ containing a disk ${\cal D}_{0}$ , and of a family of domains ${\cal D}_{t}$ converging to ${\cal D}_{0}$ as $t \to 0$ , such that the heights u t of capillary surfaces in vertical tubes with the sections ${\cal D}_{t}$ in a gravity field g satisfy $\mathop{\lim}\limits_{g \to 0} \left\{{\mathop{\inf}\limits_{{\cal D}_t} u^1- \mathop{\sup}\limits_{{\cal D}_t} u^t} \right\} = \infty$ for every $t \in (0,1)$ , but for which u 1< u 0 over ${\cal D}_{0}$ for all g > 0. In subsequent work, Finn and Lee characterized the most general convex ${\cal D}_{1}$ that leads to such a discontinuous transition when ${\cal D}_{0}$ is a disk. It has been suggested that the cause for this curious behavior is related to the fact that in all cases considered the boundaries of the ${\cal D}_{t}$ have a discontinuity in their curvatures, that is bounded below in magnitude. In the present note we present an alternative form of the example, in which the domains ${\cal D}_{t}$ are disks concentric to ${\cal D}_{0}$ . Thus, the limited smoothness in the original example of the convergence to ${\cal D}_{0}$ of the approxim ating domains cannot be viewed as the root cause of the anomaly. The procedure presented here leads to explicit bounds, which were not available in the earlier forms of the example.


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