TABLE OF CONTENTS In Ch. II the existence of solutions to the variational problem was established using the Lax Milgram theorem. There the symmetry and the ellipticity of the bilinear form a( , ) were essential hypotheses. In order to treat saddle point problems, we need a more general approach which does not require that the quadratic form be positive definite.
The relation of the functional ? to the right-hand side f of the differential equation was discussed only briefly in Remark II.4.1. A more advanced consideration of linear functionals on Sobolev spaces is appropriate here. It leads us to the so-called negative norms.
Let V be a Hilbert space. By the Riesz representation theorem, every continuous linear functional ? ? V ? can be identified with an element from V itself. Thus, often it is not necessary to distinguish between V and V ?. However, in the variational calculus, this can obscure certain important aspects of the problem. Before discussing methods of functional analysis, we first orient ourselves with a simple example.
Consider the Helmholtz equation
with f ? L 2( ?). The weak solution 
is characterized by
where ( , ) 1 is the scalar product on the Hilbert space H 1( ?). The problem (3.1) can thus be formulated as follows: Given f, find the Riesz representation for the functional
If we identify with its dual, then there is nothing...
