Collapsing and the convex hull property in a soap film capillarity model

Abstract

Soap films hanging from a wire frame are studied in the framework of capillarity theory. Minimizers in the corresponding variational problem are known to consist of positive volume regions with boundaries of constant mean curvature/pressure, possibly connected by “collapsed” minimal surfaces. We prove here that collapsing only occurs if the mean curvature/pressure of the bulky regions is negative, and that, when this last property holds, the whole soap film lies in the convex hull of its boundary wire frame.

Plateau's Problem as a Singular Limit of Capillarity Problems

“…This is of course a very large set of problems, which will require further investigations. In the companion paper [37], we start this kind of study by proving that collapsed minimizers have nonpositive Lagrange multipliers, deduce from this property that they satisfy the convex hull property, and lay the ground for the forthcoming paper [38], where we further investigate the regularity of the collapsed set K n @ £ E.…”

Plateau's Problem as a Singular Limit of Capillarity Problems

“…This is of course a very large set of problems, which will require further investigations. In the companion paper [37], we start this kind of study by proving that collapsed minimizers have nonpositive Lagrange multipliers, deduce from this property that they satisfy the convex hull property, and lay the ground for the forthcoming paper [38], where we further investigate the regularity of the collapsed set K n @ £ E.…”