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We study certain multilayer free-boundary problems, in which the layer interfaces constitute a nested family of convex closed surfaces, each characterized by a Bernoulli joining condition between the potentials in the neighboring layers. In this context, we develop convex variational methods based on a family of convexity-pre serving freeboundary perturbation operators, and we apply these methods in the study of the existence of convex solutions. 1. Introduction. The main purpose of this paper is to apply convex variational techniques to study the question of the existence of convex classical solutions to certain multiple-free-boundary problems arising in fluid dynamics, called multilayer fluid problems.

On the existence of convex classical solutions to multilayer fluid problems in arbitrary space dimensions