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<p<\infty 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

(more…)

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