Home > Functional Analysis, Partial Differential Equations > Regularity of the weak solution Part 1

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 ${L^2}$ norm of the derivatives of the solution ${u}$. 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

$\displaystyle \begin{cases} -\Delta u= f & \text{ in }\Omega \\ \hfill u=0 & \text{ on }\partial \Omega. \end{cases}$

whose weak variational form is

$\displaystyle \int_\Omega \nabla u \nabla v = \int_\Omega fv,\ \forall v \in H_0^1(\Omega).$

Note that in the variational form we only assume ${u \in H^1(\Omega)}$, and to be able to recover the PDE we need to use Green’s formula, which is valid if ${u \in H^2(\Omega)}$.

As a consequence, what I will try to do next is to prove that if ${u \in H^1(\Omega)}$ and ${u}$ satisfies the weak form of the considered problem then ${u \in H^2(\Omega)}$.

The plan of the study is:

• ${\Omega=\Bbb{R}^N}$;
• ${\Omega=\Bbb{R}^N_+=\{x_N \geq 0\}}$
• The general case which is subdivided in interior estimates and estimates near the boundary.

Now let’s take ${\Omega=\Bbb{R}^N}$ and consider for any ${h \in \Bbb{R}^N}$, ${h \neq 0}$ the function

$\displaystyle D_hu=\frac{1}{|h|}(\tau_hu-u), \text{ i.e. } D_h(x)=\frac{u(x+h)-u(x)}{|h|}.$

Consider ${v=D_{-h}(D_hu)}$ in the weak formulation (this is possible because ${u \in H^1}$). We obtain (by making a change of variables)

$\displaystyle \int_{\Bbb{R}^N} |\nabla D_h u|^2 = \int_{\Bbb{R}^N}fD_{-h}(D_hu),$

and thus

$\displaystyle \| | \nabla D_hu|^2 \|_{L^2} \leq \|f\|_2 \|D_{-h}(D_hu)\|_{L^2}.$

But we know that (using this proposition)

$\displaystyle \|D_{-h}v\|_2 \leq \|\nabla v\|_2, \forall v \in H^1,$

and in consequence we have

$\displaystyle \| | \nabla D_hu|^2 \|_{L^2} \leq \|f\|_2.$

In particular we have for every ${i=1..N}$

$\displaystyle \left\| D_h \frac{\partial u}{\partial x_i} \right\|_2 \leq \|f\|_2$

and as a consequence of the same proposition we get that

$\displaystyle \frac{\partial u}{\partial x_i} \in H^1 \text{ and therefore } u \in H^2(\Bbb{R}^N).$