On the curvature of free boundaries with a Bernoulli-type condition

In this paper we study the classical external Bernoulli problem set in an annular domain Omega of the plane. We focus on the curvature of the free boundary Gamma (outer component of the boundary of our domain) and establish a one-to-one correspondence between positive/negative curvature arcs of Gamma and of the curve gamma representing the data, extending a method put forward by A. Acker. Moreover we show that the positive curvature arcs on the free boundary bend less than the corresponding arcs on the inner curve, i.e. the maximum attained by the curvature is greater on gamma than on Gamma. Thus we can draw the following conclusions: the geometry of Gamma is simpler than that of gamma (an already known result); the shape of F is alleviated with respect to that of gamma. (c) 2006 Elsevier Ltd. All rights reserved.