# Ngô Quốc Anh

## July 24, 2010

### Regularity theory for integral equations

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 20:54

My purpose is to derive some regularity result concerning the following integral equation

$\displaystyle u(x) = \int_\Omega {\frac{{u(y)}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy}$

where $\Omega \subset \mathbb R^n$ is open and bounded and $0<\alpha. To this purpose, in this entry we first consider the equation

$\displaystyle u(x) = \int_\Omega {\frac{{f(y)}}{{{{\left| {x - y} \right|}^{n - \alpha }}}}dy}$

for a suitable choice of $f$.

The case $f \in L^\infty(\Omega)$. We will prove that $u \in C^{1,\beta}(\Omega)$ for any $\beta\in (0,1)$. Indeed, up to a constant factor, the first derivative of $u$ are given by

$\displaystyle {D_i}u(x) = \int_\Omega {\frac{{{x_i} - {y_i}}}{{{{\left| {x - y} \right|}^{n + 2 - \alpha }}}}f(y)dy}$.

From this formula,

$\displaystyle\left| {{D_i}u({x^1}) - {D_i}u({x^2})} \right| = \mathop {\sup }\limits_\Omega |f|\int_\Omega {\left| {\frac{{x_i^1 - {y_i}}}{{{{\left| {{x^1} - y} \right|}^{n + 2 - \alpha }}}} - \frac{{x_i^2 - {y_i}}}{{{{\left| {{x^2} - y} \right|}^{n + 2 - \alpha }}}}} \right|dy}$.

By the intermediate value theorem, on the line from $x^1$ to $x^2$, there exists some $x^3$ with

$\displaystyle\left| {\frac{{x_i^1 - {y_i}}}{{{{\left| {{x^1} - y} \right|}^{n + 2 - \alpha }}}} - \frac{{x_i^2 - {y_i}}}{{{{\left| {{x^2} - y} \right|}^{n + 2 - \alpha }}}}} \right| \leqslant \frac{C}{{{{\left| {{x^3} - y} \right|}^{n + 2 - \alpha }}}}\left| {{x^1} - {x^2}} \right|$.

## July 21, 2010

### Regularity Lifting by Contracting Operators

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 4:47

I just read this method in a book due to W.X. Chen and C.M. Li published by American Institute of Mathematical Sciences this year 2010.

Let $V$ be a topological vector space. Suppose there are two extended norms (i.e. the norm of an element in $V$ might be infinity) defined on $V$

$\|\cdot\|_X, \|\cdot\|_Y :V\to [0,\infty]$.

Let

$X:=\{v \in V: \|v\|_X <\infty\}$

and

$Y:=\{v \in V: \|v\|_Y <\infty\}$.

Theorem (Regularity Lifting I). Let $T$ be a contraction map from $X$ into itself and from $Y$ into itself. Assume that $f \in X$, and that there exits a function $g \in Z := X \cap Y$ such that $f = Tf + g$ in $X$. Then $f$ also belongs to $Z$.

Proof. Firstly, let

$\sqrt{\|\cdot\|_X^2+\|\cdot\|_Y^2}$

be a norm on $Z$. We first show that $T : Z \to Z$ is a contraction. Since $T$ is a contraction on $X$, there exists a constant $\theta_1$, $0 < \theta_1 < 1$ such that

$\displaystyle {\left\| {T{h_1} - T{h_2}} \right\|_X} \leqslant {\theta _1}{\left\| {{h_1} - {h_2}} \right\|_X},\quad\forall {h_1},{h_2} \in X$.

Similarly, we can find a constant $\theta_2$, $0 < \theta_2 < 1$ such that

$\displaystyle {\left\| {T{h_1} - T{h_2}} \right\|_Y} \leqslant {\theta _2}{\left\| {{h_1} - {h_2}} \right\|_Y},\quad\forall {h_1},{h_2} \in Y$.

Let $\theta = \max\{\theta_1,\theta_2\}$. Then, for any $h_1, h_2 \in Z$

$\displaystyle {\left\| {T{h_1} - T{h_2}} \right\|_Z} = \sqrt {\left\| {T{h_1} - T{h_2}} \right\|_X^2 + \left\| {T{h_1} - T{h_2}} \right\|_Y^2} \leqslant \theta {\left\| {{h_1} - {h_2}} \right\|_Z}$.

Since $T : Z \to Z$ is a contraction, given $g \in Z$, we can find a solution $h \in Z$ such that $h = Th + g$. Notice that $T : X \to X$ is also a contraction and $g \in Z \subset X$, the solution of the equation $x = Tx + g$ must be unique in $X$. Because both $h$ and $f$ are solutions of the same equation $x = Tx + g$ in $X$, we deduce that $f = h \in Z$.

Remark. In practice, we usually choose $V$ to be the space of distributions, and $X$ and $Y$ to be function spaces, for instance, $X = L^p(\Omega)$ and $Y = W^{1,q}(\Omega)$. We start from a function $f$ in a lower regularity space $X$, if we can show that $T$ is a contraction from $X$ to itself and from $Y$ to itself, then we can lift the regularity of $f$ to be in

$Z = X \cap Y = L^p(\Omega)\cap W^{1,q}(\Omega)$.

Applications to PDEs. Now, we explain how the Regularity Lifting Theorem proved in the previous subsection can be used to boost the regularity of week solutions involving critical exponent

$\displaystyle -\Delta u =u^\frac{n+2}{n-2}$.

Still assume that ­ $\Omega$ is a smooth bounded domain in $\mathbb R^n$ with $n \geqslant 3$. Let
$u \in H^1_0(\Omega)$ be a weak solution of the above PDE. Then by Sobolev embedding

$\displaystyle u \in L^\frac{2n}{n-2}(\Omega)$.

We can split the right hand side of the PDE in two parts

$\displaystyle u^\frac{n+2}{n-2}=u^\frac{4}{n-2}u=a(x)u$.

Then obviously $a \in L^\frac{n}{2}(\Omega)$. Hence, more generally, we consider the regularity of the weak solution of the following equation

$\displaystyle -\Delta u =a(x)u+b(x)$.

Theorem. Assume that $a,b \in L^\frac{n}{2}(\Omega)$. Let $u \in H_0^1(\Omega)$ be any weak solution of the foregoing PDE. Then $u \in L^p(\Omega)$ for any $1 \leqslant p<\infty$.

The proof of this theorem can be found in the book mentioned above.

Let us go back to the first PDE. Assume that $u$ is a $H^1_0(\Omega)$ weak solution. From the above theorem, we first conclude that $u$ is in $L^q(\Omega)$ for any $1 < q < \infty$. Then by a standard regularity result known as the $W^{2,p}$-regularity for a second order uniformly elliptic operator in divergence form, $u$ is in $W^{2,q}(\Omega)$. This implies that $u \in C^{1,\alpha}(\Omega)$ via Sobolev embedding. Finally, by repeated applications of the Schauder estimates, we derive that $u \in C^\infty(\Omega)$.