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