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

with

.

Thus

.

Applying the Sobolev embedding theorem again gives us

with

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.

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