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.
Acker, A., Henrot, A., Poghosyan, M., & Shahgholian, H. (2004). The multi-layer free boundary problem for the p-Laplacian in convex domains. Interfaces and Free Boundaries, 6(1), 81–103. https://doi.org/10.4171/IFB/92