The aim of this paper is to establish uniform estimates of the bottom of the spectrum of the Neumann realization of <img src="http:/fbpe/img/cubo/v12n1/img9.jpg" width="113" height="14"> on a bounded open set <img src="http:/fbpe/img/cubo/v12n1/img11.jpg" width="14" height="15">with smooth boundary when<img src="http:/fbpe/img/cubo/v12n1/img10.jpg" width="210" height="18">. This problem was motivated by a question occurring in the theory of liquid crystals and appears also in superconductivity questions in large domains.