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A subsidiary variational problem and existence criteria for capillary surfaces. (English) Zbl 0542.49021

Let Z be a cylinder of homogeneous material, closed at one end by a base of general section \(\Omega\) and partly filled with liquid. In the absence of gravity, the associated potential energy is given by \(E=\sigma \{S- \beta S^*+2HV\}\) where S is the free surface fluid area, \(\sigma\) the surface tension, \(S^*\) the area of wetted surface on Z, \(\sigma \beta\) the adhesion coefficient of fluid to cylinder and 2\(\sigma\) H a Lagrange multiplier arising from a volume constraint. The question of determining a priori conditions on \(\Omega\) for the existence of minimizing configurations was reduced by the author to a study of the functional \[ \phi(\Gamma,\gamma)=\Gamma -\Sigma^*\quad \cos \gamma +((\Sigma /\Omega)\cos \gamma)\Omega^* \] defined over curves \(\Gamma\) that cut a subregion \(\Omega^*\) from \(\Omega\) and subarc \(\Sigma^*\) from \(\Sigma\).
In the present paper, a minimizing configuration \(\Gamma\) is shown to consist, at least when \(\gamma>0\), of at most a finite number of isolated arcs. No arc \(\Gamma\) in such a configuration can enter a corner whose opening angle is less then \(\pi\). The second variation of \(\Phi\) allows to establish geometric criteria that suffice for nonexistence of a minimizing extremal \(\Gamma\) in \(\Omega\) and hence for the existence of a solution to the original problem (nonexistence-existence principle).
Reviewer: U.Massari

MSC:

49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
76T99 Multiphase and multicomponent flows
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