Today, we shall discuss a very strong tool in the theory of elliptic PDEs in order to achieve the smoothness of solution. The tool we just mentioned is known as the Calderón-Zygmund estimates or the Calderón-Zygmund inequality. Precisely,
Theorem (Calderón-Zygmund). Let and ( is open and bounded). Let be the weak solution of the following PDE
Then for any .
Let us consider the regularity of solution of
with a smooth . We also require that is bounded.
Motivation. The above PDE occurs as the Euler-Lagrange equation of the variational problem
with a smooth with is bounded and bounded away from zero. Moreover, is bounded.
In fact, to derive the Euler-Lagrange equation, we consider
where . In that case
after integrating by parts and assuming for the moment . Thus, the minimizer will verify
which turns out to be
In order to apply the -theory, we assume that is a weak solution of the PDE with
Step 1. Since is bounded, we have
Since verifies the PDE, we get
By the Sobolev embedding theorem
Applying the Sobolev embedding theorem again gives us
and . Iterating this procedure, we eventually obtain
for any .
Step 2. We now differentiate the PDE in order to obtain an equation for for all . In fact, we get
This time, we put
Since for any , it is clear to see that for any . Thus
for any which is equivalent to
for any .
Step 3. By repeating step 2 we finally get
for any and . As a consequence of the Sobolev embedding theorem, is smooth.