Local existence and uniqueness theorems for a free boundary problem

Abstract

AbstractFree boundary problems are considered, where the tangential and normal components ut and un of an otherwise unknown plane harmonic vector field are prescribed along the unknown boundary curve as a function of the coordinates x, y and the tangent angle θ. The vector field is required to exist either in the interior region G+ or in the exterior G−. In each case the free boundary is characterized by a nonlinear integral equation. A linearised version of this equation is a one‐dimensional singular integral equation. Under rather general hypotheses which are easy to check, the properties of the linear equation are described by Noether's theorems. The regularity of the solution is studied and the effect of the nonlinear terms is estimated. A variant of the Nash‐Moser implicit‐function theorem can be applied. This yields local existence and uniqueness theorems for the free boundary problem in Hölder‐classes H2+μ. The boundary curve depends continuously on the defining data. Finally some examples are given, where the linearised equation can be completely discussed.