For every compact set K⊂R^2, let operatornamecap_log(K) denote its logarithmic capacity and let operatornameCap_N(K) denote the Newtonian (electrostatic) capacity of K regarded as a subset of the plane in R^3. If D⊂R^2 is a disk satisfying operatornamecap_log(D)=operatornamecap_log(K), then operatornameCap_N(K)≤operatornameCap_N(D), with equality only when K is a disk, up to translation.

Pólya–Szegő capacity conjecture

For every compact set $K\subset\mathbb{R}^{2}$ , let $\operatorname{cap}_{\log}(K)$ denote its logarithmic capacity and let $\operatorname{Cap}_{N}(K)$ denote the Newtonian (electrostatic) capacity of $K$ regarded as a subset of the plane in $\mathbb{R}^{3}$ . If $D\subset\mathbb{R}^{2}$ is a disk satisfying $\operatorname{cap}_{\log}(D)=\operatorname{cap}_{\log}(K)$ , then $\operatorname{Cap}_{N}(K)\leq\operatorname{Cap}_{N}(D)$ , with equality only when $K$ is a disk, up to translation.

References

Primary source

arXiv

Additional references

Toward Pólya and Szegő's conjecture for logarithmic vs Newtonian capacity — arXiv — Carrie Clark, Richard S. Laugesen

geometric-analysis potential-theory

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A September 2026 paper narrows the open capacity conjecture to a stronger analytic inequality but does not prove it.

The Pólya–Szegő capacity conjecture asserts that the disk minimizes the relevant planar capacity energy, with equality only for a disk. It remains open globally.

Known results

The conjecture is proved for several perturbation classes of the disk and along certain deformations, but these are local results rather than a global proof.
Within planar sets, the disk is the only minimizer for the established lower bound; the conjectured sharp inequality remains unresolved.

September 2026 conditional reduction

Carrie Clark and Richard S. Laugesen show that the conjecture would follow from ball-minimization of a transverse strip Dirichlet energy. They also prove related $L^p$ extremality by symmetrization. This is a claimed reduction, not a proof of the required strip-energy inequality, so the main conjecture remains open.

Current status (as of September 2026): The planar capacity conjecture remains open; a new conditional reduction and related symmetrization results constitute claimed progress, not a verified solution.

Sources

No solutions have been posted yet.