A free boundary problem for the p-Laplacian: uniqueness, convexity, and successive approximation of solutions - A. Acker; R. Meyer (Electronic Journal of Differential Equations, 1995)
classical solution of a Bernoulli-type free boundary problem for the $p$-Laplace equation, $ 1 < p < infty $. In addition, we prove the existence of a classical solution in $N$ dimensions when $p = 2$ and, for $ 1 < p < infty $, results on uniqueness and starlikeness of the free boundary and continuous dependence on the fixed boundary and on the free boundary data. Finally, as an application of the trial free boundary method, we prove (also for $ 1 < p < infty $) that the free boundary is convex when the fixed boundary is convex.