Weak Formulation of Elliptic BVP’s

There are a large number of problems of physical interest that can be formulated in the abstract setting in which the Lax-Milgram lemma is applied to an equation expressed in terms of an appropriate bilinear form to prove existence of a unique solution. The selection of the solution space and the bilinear form controls the PDE and boundary conditions that the solution will satisfy. We will give some examples to illustrate the interplay between the bilinear form, the solution space and the formal problem that the corresponding solution satisfies.