×

A free boundary problem for the \(p\)-Laplacian: uniqueness, convexity, and successive approximation of solutions. (English) Zbl 0820.35037

Summary: We prove convergence of a trial free boundary method to a 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.

MSC:

35J20 Variational methods for second-order elliptic equations
35A35 Theoretical approximation in context of PDEs
35R35 Free boundary problems for PDEs
PDFBibTeX XMLCite
Full Text: EuDML EMIS