A proof of the invariance of the contact angle in Electrowetting

We prove the invariance of the contact angle in liquid-solid wetting phenomena : an electrified droplet is spreading on a solid surface. The drop is minimizing 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 method, using well-chosen test functions, we obtain the main result about the contact angle between the drop and the solid.