# How does electric potential relate to mean curvature?

Consider a compact, convex domain $\Omega \subset \mathbb{R}^3$ with $|\Omega|=1$ with smooth boundary $\partial \Omega$.

Now consider the electric potential generated by this uniform mass distribution: $\phi = \int_{\Omega} \frac{1}{|x-y|} dx$. Question: I would like to know if there is a relationship between mean curvature $H$ and $\phi$. What I have in mind is an inequality of the form $\|\phi - \bar \phi\|_{L^p(\partial \Omega)} \leq C \|H - \bar H\|_{L^p(\partial \Omega)}$ where $\bar \phi$ and $\bar H$ denote the averages over $\partial \Omega$ of $\phi$ and $H$ respectively and $p$ can be anything for the time being.

Although one term is much more 'non-local' than the other, for a convex body it seems reasonable that the closer to being an surface of constant mean curvature, the closer one is to being an equipotential surface. This is related to a previous question of mine which ended up going unanswered regarding whether the only convex, compact equipotential surface in $\mathbb{R}^3$ was a sphere or not.

• I have no time to look for a precise reference, but this belongs to the general theory of rearrangement in PDEs. Reliable authors are Talenti, Bandle and perhaps Mossino. Sep 8, 2011 at 15:46
• Isn't $\phi$ infinite along the boundary of $\Omega$? Sep 8, 2011 at 16:11
• No. It's like integrating 1/r in R^3 which will give you something of order 1 since surface measure is r^2 Sep 8, 2011 at 16:40
• A more elaborate way of seeing this is that the characteristic function of $\Omega$ is an $L^p$ function for all $p \in [1,\infty]$ and standard elliptic theory says that $\phi$ must be $H^2$ so in fact continuous. Sep 8, 2011 at 16:42
• Isn't this problem related to the capacitor problem? Have a look at Sections 11.15-11.17 of the book "Analysis" of E. H. Lieb and M. Loss (2nd. edition), or the book "Function Spaces and Potential Theory" by D. R. Adams and L. I. Hedberg. Sep 8, 2011 at 21:23

This seems very related to the Eshelby / Polya-Szegö conjecture, which asks (in one of its formulations, see Liu) whether the fact that the solutions of $$\Delta \phi = 1_\Omega$$ (plus appropriate decay at infinity) are of the form $$\sum_{i=1}^d a_k x_k^d +C \textrm{ in } \Omega$$ implies that the $\Omega$ is a ball, or an ellipsoid. This property is true in dimension 2 and 3, but it is also known to be extremely unstable, see e.g. Kang.