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.

Lemma 1. For every x \in \mathbb R^n, there exists \lambda_0 (x ) > 0 such that

u_{x,\lambda}(y ) < u(y ),

for all 0 < \lambda < \lambda_0(x) and |y - x| > \lambda.

Apparently, Lemma 1 has its own geometric meaning as can be seen from the form of solution u to the PDE, i.e. u is in fact symmetric and monotone decreasing. The proof of Lemma 1 is mainly based on the maximum principle. Set, for x \in \mathbb R^n,

\displaystyle\overline \lambda (x) = \sup \left\{ {\mu > 0:{u_{x,\lambda }}(y) \leqslant u(y) \quad \forall \left| {y - x} \right| \geqslant \lambda , \quad 0 < \lambda \leqslant \mu } \right\}.

By the lemma above, \overline \lambda(x) is well-defined and 0 < \overline \lambda(x) \leqslant \infty for x \in \mathbb R^n.

Lemma 2. If \overline \lambda (x)<\infty for some x \in \mathbb R^n then u_{x,\overline \lambda (x)}\equiv u on \mathbb R^n\backslash\{x\}.

Lemma 2 is proved using contradiction argument and the Hopf lemma.

Lemma 3. If \overline \lambda (x)=\infty for some \overline x \in \mathbb R^n then \overline \lambda (x)=\infty for all x \in \mathbb R^n.

The proof of Lemma 3 is based on the following asymptotic behavior

\displaystyle\mathop {\lim }\limits_{|y| \to \infty } |y{|^{n - 2}}u(y) = \infty .

Lemma 4. \overline \lambda (x)<\infty for all x \in \mathbb R^n.

This lemma can be proved using contradiction argument and making use of the first fundamental lemma. In fact, by contradiction if this is the case then u is a constant which can not be true.

Proof of theorem. It follows from Lemma 2 and Lemma 4 that for every x \in \mathbb R^n, there exists \overline\lambda(x) > 0 such that

u_{x,\overline \lambda (x)}\equiv u.

Then by the second fundamental lemma in this entry, for some a, d > 0 and some \overline x \in \mathbb R^n,

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

The theorem follows from the above and the fact that u is indeed a solution of the PDE.

Remark. The method of moving spheres captures the solutions directly rather than going through the usual procedure of proving radial symmetry of solutions and then classifying radial solutions.

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