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<n. 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|.

(more…)

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