# Ngô Quốc Anh

## August 29, 2010

### Achieving regularity results via bootstrap argument, 4

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 2:36

Let us consider the following equation

$\displaystyle u(x) = \int_{{\mathbb{R}^n}} {\frac{{u{{(y)}^{\frac{{n + \alpha }}{{n - \alpha }}}}}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy} , \quad x \in {\mathbb{R}^n}$

for $n\geqslant 1$ and $0<\alpha. In this entry, by using boothstrap argument, we show that

Theorem. If positive function $u \in L_{loc}^\frac{2n}{n-\alpha}(\mathbb R^n)$ solves the equation, then $u \in C^\infty(\mathbb R^n)$.

In the process of proving the result, we need the following result

Proposition. Let $V \in L^\frac{n}{\alpha}(B_3)$ be a non-negative function and set

$\displaystyle \delta(V)=\|V\|_{L^\frac{n}{\alpha}(B_3)}$.

For $\nu >r>\frac{n}{n-\alpha}$, there exist positive constants $\overline \delta<1$ and $C \geqslant 1$ depending only on $n, \alpha, r$ and $\nu$ such that for any $0 \leqslant V \in L^\frac{n}{\alpha}(B_3)$ with $\delta(V) \leqslant \overline \delta$, $h \in L^\nu(B_2)$ and $0 \leqslant u \in L^r(B_3)$ satisfying

$\displaystyle u(x) \leqslant \int_{{B_3}} {\frac{{V(y)u(y)}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy} + h(x), \quad x \in {B_2}$

we have

$\displaystyle {\left\| u \right\|_{{L^\nu }({B_{1/2}})}} \leqslant C\left( {{{\left\| u \right\|}_{{L^r}({B_3})}} + {{\left\| h \right\|}_{{L^\nu }({B_2})}}} \right)$.

## 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

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

## May 8, 2010

### Achieving regularity results via bootstrap argument

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 2:35

Let us now consider a very important technique in PDEs called the  bootstrap method or bootstrap argument. Actually, this method is a part of the proof of the Schauder estimates, etc. I will follow the recent paper due to Li, Strohmer and Wang published in Proc. Amer. Math. Soc. in 2009.

The equation considered here is the following

$\displaystyle u(x) = A\int_\Omega {\frac{{|u|^p (y)}}{{|x - y{|^{n - \alpha }}}}dy} + B$

together with

$u\big|_{\partial\Omega}=\beta$

where $\Omega$ is a $C^1$ bounded domain. We also assume

$p,A >0, \quad \beta, B \geqslant 0, \quad 1<\alpha .

Theorem. If $u \in L^q(\Omega)$ is a solution to the PDE for some $q>1$ then $u \in C(\overline \Omega)$.

In order to run the bootstrap argument, we need the following auxiliary result

Lemma. Suppose $w\in L^r(\Omega)$ with $1\leqslant r<\infty$ and

$\displaystyle v(x) = A\int_\Omega {\frac{{|w| (y)}}{{|x - y{|^{n - \alpha }}}}dy} + B$

then $v \in W^{1,s}(\Omega)$ where

$\begin{cases}\frac{1}{s}+\frac{\alpha-1}{n}=\frac{1}{r}, & {\rm if }\; 1 \leqslant r <\frac{n}{\alpha-1},\\{\rm any} \; s \geqslant 1, & {\rm if } \; r \geqslant \frac{n}{\alpha-1}.\end{cases}$