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The main result of this paper concerns existence of classical solutions to the multi-layer Bernoulli free boundary problem with nonlinear joining conditions and the p-Laplacian as governing operator. The present treatment of the two-layer case involves technical refinements of the one-layer case, studied earlier by two of the authors. The existence treatment of the multi-layer case is largely based on a reduction to the two-layer case, in which uniform separation of the free boundaries plays a key role.

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In this paper we study a free boundary problem for the heat equation in a convex ring. Here we prove that under some conditions on initial data, the considered problem has unique solution.