Ball minimization of transverse Neumann-strip energy

Prove that, for compact sets in the central hyperplane of an infinite horizontal strip and their equilibrium potentials satisfying the stated Neumann boundary conditions and decay estimates, the closed ball minimizes the vertical component of the Dirichlet energy, with equality only for a translate of the ball up to a set of inner \((n-1)\)-capacity zero.

Background

The conjecture compares an equilibrium potential $U$ associated with an arbitrary compact set $K\subset\mathbb{R}^n$ and a potential $V$ associated with a closed ball $B$, both in a strip with Neumann conditions on the horizontal boundaries.

Its conclusion is the inequality $\int_S(\partial_zU)^2\geq\int_S(\partial_zV)^2$. The paper’s main theorem shows that this conjecture implies the higher-dimensional capacity-ratio conjecture, while the paper only establishes supporting $L^p$-symmetrization inequalities.