Ngô Quốc Anh

May 10, 2010

The method of moving spheres: An introduction

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 15:16

The method of moving spheres is a variant of the method of moving planes. Roughly speaking, one makes reflection with respect to spheres instead of planes, and then obtain the symmetry of solutions.

Let us consider the following PDE

$\displaystyle -\Delta u = n(n-2)u^\frac{n+2}{n-2}, \quad u>0$

in $\mathbb R^n$, $n \geqslant 3$.

The classification of positive solutions of the above PDE had been done by Caffarelli, Gidas and Spruck [here] around 1989.

Theorem (Caffarelli, Gidas and Spruck).  A $C^2$ solution of the PDE above is of the form

$\displaystyle u(x) = {\left( {\frac{a}{{d + {{\left| {x - \overline x } \right|}^2}}}} \right)^{\frac{{n - 2}}{2}}}$.

It is worth noticing that under the additional hypothesis on the asymptotic behavior of $u(x)$, the result was established earlier by Obata [here] and Gidas, Ni and Nirenberg [here]. The proof of Obata is more geometric, while the proof of Gidas, Ni and Nirenberg is by the method of moving planes. The proof of Caffarelli, Gidas and Spruck is by a “measure theoretic” variation of the method of moving planes.

Such Liouville-type theorems have played a fundamental role in the study of semilinear elliptic equations with critical exponent, which include the Yamabe problem (prescribing scalar curvature) and the Nirenberg problem (prescribing Gaussian curvature problem).

Around 2003, Li and Zang [here] gave a different proof of the above theorem. Instead of proving the radial symmetry of any solution $u$ and then deducing the explicit shape of it by ODE methods (as making use of the method of moving planes), the authors here derive the form of the solution directly using the method of moving spheres which turns out to be the main point of this entry.

For the sake of simplicity, let us briefly stress several points of the proof. For $x \in \mathbb R^n$ and $\lambda > 0$, consider the Kelvin transform of $u$ with respect to a ball centered at $x$ with radius $\lambda$

$\displaystyle {u_{x,\lambda }}(y) = \frac{{{\lambda ^{n - 2}}}}{{{{\left| {y - x} \right|}^{n - 2}}}}u\left( {x + {\lambda ^2}\frac{{y - x}}{{{{\left| {y - x} \right|}^2}}}} \right), \quad \forall y \in {\mathbb{R}^n}\backslash \left\{ x \right\}$.

This Kelvin transform is actually sort of reflection of function. Obviously if $y$ lies within the ball $B_\lambda(x)$ then $x+\lambda^2 \frac{y-x}{|y-x|^2}$ lies outside the ball and vise versa. The proof was divided into several lemmas. Our first lemma says that the method of moving spheres can get started.