On the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions

Acker, Andrew

doi:10.1090/S0002-9947-98-01943-6

On the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions
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Trans. Amer. Math. Soc. 350 (1998), 2981-3020 Request permission

Abstract:

We prove the existence of convex classical solutions for a general multidimensional, multilayer free-boundary problem. The geometric context of this problem is a nested family of closed, convex surfaces. Except for the innermost and outermost surfaces, which are given, these surfaces are interpreted as unknown layer-interfaces, where the layers are the bounded annular domains between them. Each unknown interface is characterized by a quite general nonlinear equation, called a joining condition, which relates the first derivatives (along the interface) of the capacitary potentials in the two adjoining layers, as well as the spatial variables. A well-known special case of this problem involves several stationary, immiscible, two-dimensional flows of ideal fluid, related along their interfaces by Bernoulli’s law.

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