A proof of the invariance of the contact angle in electrowetting

We prove the invariance of the contact angle in liquid–solid electrowetting phenomena: an electrified droplet is spreading on a solid surface. The drop minimizes its energy. We express the differential of this energy with respect to the shape of the drop and deduce necessary conditions for optimality. By variational methods, using well-chosen test functions, we obtain the main result about the contact angle between the drop and the solid.