Singular Free Boundary Problems and Local Bifurcation Theory

A constructive method applicable to the solution of a wide class of free boundary problems is presented. A solution-dependent transformation technique is introduced. By considering a singular limit of the transformation, a related problem, to which local bifurcation theory may be applied, is derived. By inverting the (near singular) mapping between the two problems, an expression for solutions of the original problem is obtained. The method is illustrated by the study of a singularly perturbed elliptic equation. Approximate solutions are constructed and the validity of the approximations established by means of the Contraction Mapping Theorem.