Ngô Quốc Anh

June 1, 2010

The method of moving spheres: An integral form

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 1:05

Let us now consider the following equation

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

where \alpha is a real number satisfying 0<\alpha<n.

Lieb [here] proved among other things that there exist maximizing functions f for the Hardy-Littlewoord-Sobolev inequality on \mathbb R^n

\displaystyle {\left\| {\int_{{\mathbb{R}^n}} {\frac{{f(y)}}{{{{\left| { \cdot - y} \right|}^\lambda }}}dy} } \right\|_{{L^q}({\mathbb{R}^n})}} \leqslant {C_{p,\lambda ,n}}{\left\| f \right\|_{{L^p}({\mathbb{R}^n})}}

When p=\frac{2n}{n-\lambda} and q=\frac{2n}{\lambda}, the Euler-Langrange equation for a maximizing f is nothing but our integral equation.

Having discussion of fractional Laplacian [here] one can easily see that 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}.

The classification of solution to the integral equation was done by Chen, Li and Ou [here] published in Comm. Pure App. Math. around 2006 via the integral form of the method of moving planes. Our goal is to derive a new approach based on the integral form of the method of moving spheres. This result was due to Zhang and Hao [here] published in J. Math. Anal. Appl. around 2008.

Theorem. Let u \in L_{loc}^\frac{2n}{n-\alpha}(\mathbb R^n) be a positive solution to the integral equation. Then u(x) is radially symmetric and has the form

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

for some constants a,d>0 and \overline x \in \mathbb R^n.

Outline of the proof. Let v be a positive function on \mathbb R^n, for x \in \mathbb R^n and \lambda>0 we define

\displaystyle {v_{x,\lambda }}(y) = {\left( {\frac{\lambda }{{\left| {y - x} \right|}}} \right)^{n - \alpha }}v({y^{x,\lambda }}), \quad y \in {\mathbb{R}^n}

where

\displaystyle {y^{x,\lambda }} = x + {\lambda ^2}\frac{{y - x}}{{{{\left| {y - x} \right|}^2}}}.

Set

\displaystyle \tau = \frac{n+\alpha}{n-\alpha}.

Lemma 1. For any solution u of the integral equation, we have

\displaystyle {u_{x,\lambda }}(y) - u(y) = \int\limits_{\left| {y - x} \right| \geqslant \lambda } {\left[ {\frac{1}{{{{\left| {y - z} \right|}^{n - \alpha }}}} - {{\left( {\frac{\lambda }{{\left| {y - x} \right|}}} \right)}^{n - \alpha }}\frac{1}{{\left| {{y^{x,\lambda }} - z} \right|^{n - \alpha }}}} \right]\left( {{u_{x,\lambda }}{{(z)}^\tau } - u{{(z)}^\tau }} \right)dz}.

This lemma has the same form of the lemma considered in this entry. In fact, the proof is straightforward.

Lemma 2. For x \in \mathbb R^n, there exists \lambda_0(x)>0 such that

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

for any 0<\lambda<\lambda_0(x) and |y-x| \geqslant \lambda.

This lemma tells us that we can run the method of moving spheres. The idea is as follows: it follows from the form of our solution that our solution is indeed monotone decreasing. Thus starting from a point x it must be possible to find such a \lambda_0 so that outside a sphere centered at x with suitable radius, the attitude of function is lower that that at x_0. The proof is similar to the proof of step 1 in this entry.

Next for each x\in \mathbb R^n, we define

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

Lemma 3. If \overline \lambda (x_0) < \infty for some x_0 \in \mathbb R^n then

\displaystyle u_{x_0,\overline \lambda (x_0)} \equiv u

in \mathbb R^n.

The proof of this lemma is similar to the proof of step 2 in this entry. We do it by contradiction argument. Having all discussion above, we are able to complete the proof of theorem. In fact, w shall prove that \overline\lambda(x) is finite for all x.

Proof of theorem. If there exists some x_0 such that \overline \lambda (x_0) < \infty, then by Lemma 3,

\displaystyle\mathop {\lim }\limits_{|y| \to \infty } {\left| y \right|^{n - \alpha }}u(y) = \overline \lambda {({x_0})^{n - \alpha }}u({x_0}) < \infty.

By the definition of \overline \lambda,

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

Multiply the above by |y|^{n-\alpha} and let |y| \to \infty we get

\displaystyle\mathop {\lim \inf }\limits_{|y| \to \infty } {\left| y \right|^{n - \alpha }}u(y) \geqslant {\lambda ^{n - \alpha }}u(x),\forall 0 < \lambda < \overline \lambda (x).

Thus

\overline\lambda(x)<\infty, \quad \forall x.

It now follows from the Lemma 3 that

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

for any x. Thus gives

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

by the second fundamental lemma [here]. If \overline \lambda(x)=\infty for any x then

\displaystyle {{u_{x,\lambda }}(y) \leqslant u(y), \quad \forall \left| {y - x} \right| \geqslant \lambda>0 }.

It now follows from the first fundamental lemma [here] that u is constant which contradiction with the integral equation.

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.

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May 3, 2010

The second fundamental lemma in the method of moving spheres

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 3:32

Today we continue to discuss the second fundamental lemma in the method of moving spheres. This lemma also comes from a paper due to Y.Y. Li published in J. Eur. Math. Soc. (2004).

Recall from the previous entry where the first lemma was considered if the following

\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

holds then f is constant or \pm \infty. We now consider the equality case. Precisely,

Lemma. Let n\geqslant 1, \nu \in \mathbb R and f \in C^0(\mathbb R^n). Suppose that for every x \in \mathbb R^n there exists \lambda(x)>0 such that

\displaystyle {\left( {\frac{\lambda(x) }{{|y - x|}}} \right)^\nu }f\left( {x + {\lambda(x) ^2}\frac{{y - x}}{{|y - x{|^2}}}} \right) =f(y), \quad \forall |y - x| > 0.

Then for some a \geqslant 0, d>0 and \overline x \in \mathbb R^n

\displaystyle f(x) = \pm a{\left( {\frac{1}{{d + {{\left| {x - \overline x } \right|}^2}}}} \right)^{\frac{\nu }{2}}}.

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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.

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