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