Elliptic Boundary Value Problem - Weak or Variational Formulation

Weak or Variational Formulation

The first step to cast the boundary value problem as in the language of Sobolev spaces is to rephrase it in its weak form. Consider the Laplace problem . Multiply each side of the equation by a "test function" and integrate by parts using Green's theorem to obtain

.

We will be solving the Dirichlet problem, so that . For technical reasons, it is useful to assume that is taken from the same space of functions as is so we also assume that . This gets rid of the term, yielding

(*)

where

and

.

If is a general elliptic operator, the same reasoning leads to the bilinear form

.

We do not discuss the Neumann problem but note that it is analyzed in a similar way.