# 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

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