[en] In this report, the theory of constrained elliptical problems is presented. Two classes of methods are used to solve such problems, which are the dualization and the penalty method. In the first one, use is made of Lagrange multipliers. The justification of this method has been given by Brezzi in 1974. This theory is presented in the present report by a particularly simple way, where use is made of elementary hilbertian geometry. Our step, which consists in building the solution, emphasizes the natural character of Brezzi's conditions. Extending this analysis, it was rational to also examine the penalty method, which is often preferred because its simpler implementation. It is shown that the convergence of this method necessitate some hypotheses which are in most cases equivalent to Brezzi's conditions.