Ngô Quốc Anh

April 5, 2010

The first fundamental lemma in the method of moving spheres

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 21:37

I am going to discuss two fundamental lemmas appearing in the method of moving spheres. They have been used repeatedly in many works. For the full of account, the paper due to Y.Y. Li published in J. Eur. Math. Soc. (2004) is the best.

Lemma. For $n \geq 1$ and $\nu \in \mathbb R$, let $f$ be a function defined on $\mathbb R^n$ and valued in $[-\infty, +\infty]$ satisfying $\displaystyle {\left( {\frac{\lambda }{{|y - x|}}} \right)^\nu }f\left( {x + {\lambda ^2}\frac{{y - x}}{{|y - x{|^2}}}} \right) \leqslant f(y), \quad \forall |y - x| > \lambda > 0$.

Then $f$ is constant or $\pm \infty$.

Proof. For all $b>1$ and $y,z \in \mathbb R^n$ with $y \ne z$, let $\displaystyle x = x(b) = y + b(z - y),\quad \lambda = \lambda (b) = \sqrt {|z - x||y - x|}$.

Then $\displaystyle z={x + {\lambda ^2}\frac{{y - x}}{{|y - x{|^2}}}}$.

Therefore, $\displaystyle {\left( {\frac{\lambda }{{|y - x|}}} \right)^\nu }f(z) \leqslant f(y)$.

Since $\displaystyle\mathop {\lim }\limits_{b \to \infty } \frac{\lambda }{{|y - x|}} = \mathop {\lim }\limits_{b \to \infty } \sqrt {\frac{{|z - x|}}{{|y - x|}}} = 1$

we have $f(z)\leq f(y)$.

Our lemma follows since $y \ne z$ are arbitrary.

By what we have discussed in here, it is worth noticing that $z=R_{x,\lambda}(y)$, the reflection of $y$ on the sphere with center at $x$ and radius $\lambda>0$. Therefore, the lemma has its own geometric meaning. For example, when $\nu=0$, the unique function satisfying $\displaystyle f(R_{x,\lambda}(y)) \leq f(y)$

is constant or $\pm \infty$.

The second fundamental lemma will concern the case of equality. The shape of such functions plays an important role in PDEs. Having discuss these lemmas, I am going to introduce several variants of the moving spheres and their applications to PDEs.