a) Let be a constant. State what is meant by a weak solution of the PDE on .
b) Show that if is a continuous function of one real variable, then is a weak solution of the PDE on .
Let be a bounded open set with smooth boundary . Let be a smooth family of symmetric real matrices that are uniformly positive definite. Let and be smooth functions on . Define the functional
where is the scalar product on . Suppose that is a minimizer of this functional subject to the Dirichlet condition on , with continuous.
a) Show that satisfies the variational equation
for any .
b) What is the PDE satisfied by ?
c) Suppose . Show that is an admissible test function, and use this to conclude that . (Hint: You may use the fact a.e.)
d) Show that there can be only one minimizer of or, equivalently, only one solution of the corresponding PDE.
For , define . Then since minimizes the functional, . We can calculate (by exploiting the symmetry of ):
And so which proves the result.
The boundary terms vanish since and we've obtained a weak form of the PDE. Thus, is a solution to the following PDE:
First we need to show that . Firstly, on , since we've assumed . Secondly, , hence , must be Lipschitz continuous since and is a bounded domain in (i.e. ) and so must achieve a (finite) maximum in , hence the derivative is bounded, hence is Lipschitz. Therefore, .
This gives .
But notice that since is uniformly positive definite, then
Therefore, we have
a contradition, unless , i.e. a.e.
Suppose are two distinct such solution. Let . Then is Lipschitz (since must both be) and on . Therefore, the variational equation gives
. Since is positive definite, this gives
, a contradiction unless a.e.