Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I've run into the problem of trying to evaluate the following:

$\max_{ \xi} \iint_{\partial B \times \partial B} \xi(x) \Phi(|x-y|) \xi(y) dS(x)dS(y)$

subject to $\int_{\partial B} \xi(y)dS(y) = 0$ and $\int_{\partial B}\xi^2(y)dS(y)=1$ where $B \subset \mathbb{R}^3$ is a ball of radius $1$ and $\Phi(|x-y|)=\frac{1}{|x-y|}$ is the Newtonian potential.

This seems to resemble an inverse fractional Soblev norm such as $H^{-1}$ and moreover appears to be related to the problem of finding an optimal Poincare constant.

My guess is that the maximum is obtained for $\xi=+1$ on the upper half and $\xi=-1$ on the lower half. Given this however, I still cannot do an explicit calculation to determine this quantity. Is there a standard reference for such problems arising in Potential Theory perhaps which will allow one to evaluate (even approximately) such expressions?

For instance I know I can rewrite the above as: $\int_{\partial B} |\nabla w|^2$ where $-\Delta w = \mu$ and $\mu(x) = \xi(x)dS(x)$ but I'm not sure how this can help me to evaluate such an expression.

To summarize, I would like to try to evaluate the above double integral for the particular function $\xi = +1$ on the upper half of the ball and $\xi=-1$ on the lower half. Being able to solve explicitly the above maximization problem would be a bonus.

share|improve this question
    
In terms of $w$, the solution of a Dirichlet problem with finte dimensional constraints is smooth, likely analytic. Thus the optimal $\xi$ must be smooth. It cannot be a sign function. –  Denis Serre Nov 8 '11 at 7:07
    
This is not a dirichlet energy, it is a non-local H^{-1} type energy and so solutions need not be smooth. –  Dorian Nov 11 '11 at 3:14

1 Answer 1

The potential of $\xi$ is a harmonic function $u$ on the unit ball in the obvious way. $u(x) = \int_{\partial B} \frac{\xi(y)}{|x-y|} dS(y).$

Consider the spherical harmonics decomposition of $u$, given by

$u = \sum_{l=0}^\infty f^l r^l Y^l (\theta, \phi)$

The energy given above reduces to $C_n \int_{\partial B} u u_\nu dS = \sum l |f^l|^2$, while the normalization condition you have given reduces to $\int_{\partial B} |u_\nu|^2 dS = 1 = \sum l^2 |f^l|^2$. Ignoring the case $l=0$ (which has zero energy), you get the maximum of the two ratios at $l=1$ or $f^l = \delta_{1l}$.

share|improve this answer
    
So are you saying that the maximum ration is $\sum l |f^l|^2 = 1$ since $f^l=\delta_{1l}$? –  Dorian Dec 6 '11 at 20:42
    
I find that strange as it should depend on $r$ but perhaps I'm missing something from your explanation. –  Dorian Dec 6 '11 at 20:43
    
Possibly involving some constant depending on dimension, yes. I just realized there may also be a factor of 2 involved, since my expression only captures the energy on the inside of the ball (in this case, $u$ on the outside of the ball is just the Kelvin transform of $u$ on the inside). –  Ray Yang Dec 8 '11 at 15:42

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.