An $L^2$ Ahlfors--Beurling inequality in every dimension

arXiv:2610.37886v1 Announce Type: new Abstract: Let $\Omega\subset\mathbb{R}^N$ ($N\geq2$) be measurable with $0<|\Omega|<\infty$, and let $u_\Omega$ be its logarithmic potential ($N=2$) or Newtonian potential ($N\geq3$), normalized so that $\Delta u_\Omega=\chi_\Omega$. We prove the sharp inequality \[ \int_\Omega|\nabla u_\Omega|^2\,\mathrm{d} x \leq\frac{1}{2N\omega_{N-1}} \iint_{\Omega\times\Omega}\frac{\mathrm{d} x\,\mathrm{d} y}{|x-y|^{N-2}}, \] where $\omega_{N-1}=|\mathbb{S}^{N-1}|$, with equality if and only if $\Omega$ is a ball up to a null set. In dimension two this settles the $L^2$ Ahlfors--Beurling conjecture of Fleeman and Khavinson. For $N\ge3$ it is equivalent to the scale-free bound that at most the fraction $(N-2)/(2N)$ of the total field energy lies inside $\Omega$. We also obtain an explicit deficit identity, a sharp extension to bounded nonnegative densities, and consequences for torsional rigidity, Bergman analytic content, Brownian motion, and aggregation equations.

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