Ngô Quốc Anh

May 7, 2010

The method of moving planes: An integral form

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 0:41

Today we study a beautiful version of the method of moving planes. This method just invented around 2003 by W.X. Chen, C.M. Li and B. Ou and published in Comm. Pure Appl. Math. around 2006 (for the paper, please go here). In that paper, the author proved among other things that every positive regular solution u(x) of the integral equation

\displaystyle u(x) = \int_{{\mathbb{R}^n}} {\frac{1}{{{{\left| {x - y} \right|}^{n - \alpha }}}}u{{(y)}^{\frac{{n + \alpha }}{{n - \alpha }}}}dy}

is radially symmetric and monotone about some point and therefore assumes the form

\displaystyle u(x) = c{\left( {\frac{t}{{{t^2} + {{\left| {x - {x_0}} \right|}^2}}}} \right)^{\frac{{n - \alpha }}{2}}}

with some constant c=c(n,\alpha) and for some t>0 and x_0 \in \mathbb R^n. These solutions in case n=3 and \alpha=2 have the following shape.

This integral equation is also closely related to the following family of semilinear PDEs

\displaystyle {( - \Delta )^{\frac{\alpha }{2}}}u = {u^{\frac{{n + \alpha }}{{n - \alpha }}}}.


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