# Ngô Quốc Anh

## June 13, 2010

### Achieving regularity results via bootstrap argument, 2

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 1:37

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 $L^p$ estimates or the Calderón-Zygmund inequality. Precisely,

Theorem (Calderón-Zygmund). Let $1 and $f \in L^p(\Omega)$ ($\Omega$ is open and bounded). Let $u$ be the weak solution of the following PDE

$\displaystyle \Delta u = f$.

Then $u\in W^{2,p}(\Omega')$ for any $\Omega' \Subset \Omega$.

Let us consider the regularity of solution of

$\displaystyle \Delta u +\Gamma(u)|\nabla u|^2=0$

with a smooth $\Gamma$. We also require that $\Gamma$ is bounded.

Motivation. The above PDE occurs as the Euler-Lagrange equation of the variational problem

$\displaystyle I(u)=\int_\Omega g(u(x))|\nabla u(x)|^2dx \to {\rm min}$

with a smooth $g$ with is bounded and bounded away from zero. Moreover, $g'$ is bounded.

In fact, to derive the Euler-Lagrange equation, we consider

$\displaystyle I(u + t\varphi ) = \int_\Omega {g(u + t\varphi ){{\left| {\nabla (u + t\varphi )} \right|}^2}dx}$

where $\varphi \in H_0^{1,2}(\Omega)$. In that case

$\displaystyle \frac{d}{{dt}}I(u + t\varphi ) = \int_\Omega {\left[ { - 2g(u)\Delta u - g'(u){{\left| {\nabla u} \right|}^2}} \right]\varphi dx}$

after integrating by parts and assuming for the moment $u \in C^2$. Thus, the minimizer will verify

$\displaystyle - 2g(u)\Delta u - g'(u){\left| {\nabla u} \right|^2} = 0$