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

We focus on the curvature of the free boundary Γ (outer component of the boundary of our domain) and establish a one-to-one correspondence between positive/negative curvature arcs of Γ and of the curve γ 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 γ than on Γ. Thus we can draw the following conclusions: the geometry of Γ is simpler than that of γ (an already known result); the shape of Γ is alleviated with respect to that of γ.