## Regularity of the weak solution Part 1

I will present here how to recover the regularity of a weak solution for the Dirichlet problem. The arguments can easily be adapted to most of the weak formulations involving the Laplace operator, the essential tool being the estimate of the norm of the derivatives of the solution . The arguments are adapted from H. Brezis, *Functional Analysis, Sobolev Spaces and Partial Differential Equations*, Chapter 9, and the tool named the *method of translations* is due to L. Niremberg.

Consider the problem

whose weak variational form is

Note that in the variational form we only assume , and to be able to recover the PDE we need to use Green’s formula, which is valid if .

As a consequence, what I will try to do next is to prove that if and satisfies the weak form of the considered problem then .

The plan of the study is:

- ;
- The general case which is subdivided in
*interior estimates*and*estimates near the boundary*.

Now let’s take and consider for any , the function

Consider in the weak formulation (this is possible because ). We obtain (by making a change of variables)

and thus

But we know that (using this proposition)

and in consequence we have

In particular we have for every

and as a consequence of the same proposition we get that