A weak solution of the biharmonic equation,
is a function such that
for all .
Assume that is a bounded subset of with smooth boundary and use the weak formulation of the problem to prove the existence of a unique weak solution.
Consider the functional . is bilinear by linearity of the Laplacian. Now, we claim that is also continuous and coercive.
where the first inequality is due to Holder and the second is by the definition of the Sobolev norm. And so is a continuous functional.
To show coercivity, we use the fact that by two uses of integration by parts, which gives
which establishes coercivity.
Thus, by the Lax-Milgram Theorem, the weak solution exists and is unique.