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.
Read more about this topic: Elliptic Boundary Value Problem
Famous quotes containing the words weak or, weak and/or formulation:
“Necessity makes women very weak or very strong, and pent-up rivers are sometimes dangerous. Look to it!”
—Mary Worthington, U.S. womens magazine contributor. The Lily, p. 183 ( November 1856)
“Were I to buy this life of pleasure and this only chance at happiness with a few little dangers, where would be the harm? And wouldnt it still be fortunate to find a weak excuse to give her proof of my love?”
—Stendhal [Marie Henri Beyle] (17831842)
“Art is an experience, not the formulation of a problem.”
—Lindsay Anderson (b. 1923)