Existence and non existence of capillary surfaces

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A general criterion for existence of solutions of the capillary equation, introduced by Concus and Finn [1] and by Giusti [7], is shown to be equivalent to the question of existence of certain vector fields. The result is applied to particular boundary configurations, and it is shown that in some cases the local “corner condition” of [1] is both necessary and sufficient in the global configuration. In other situations a different kind of unstable dependence on the boundary geometry appears, that could not have been predicted by previous results.