Magnanini’s conjecture on critical points of the torsion function

For every bounded simply-connected domain $\Omega\subset\mathbb{R}^2$ , let $u$ be its torsion function, defined by $-\Delta u=1$ in $\Omega$ and $u=0$ on $\partial\Omega$ , and let $d(x)=\operatorname{dist}(x,\partial\Omega)$ . Magnanini's conjecture asserts that the number of local maxima of $u$ in $\Omega$ is at most the number of local maxima of $d$ in $\Omega$ ; equivalently, $N_{\mathrm{max}}(u)\leq N_{\mathrm{max}}(d)$ .

Progress summary

Solved

A new preprint reports counterexamples that would disprove the conjecture, but the result has not yet been independently verified.

Magnanini’s conjecture proposes an upper bound for the number of critical points of the torsion function in terms of local maxima of the distance function on planar domains. The conjecture is attributed to Magnanini; no originating date was identified.

August 2026 counterexamples

Edwards, Parker, Lundberg, Erik, Ramachandran, and Koushik report counterexamples disproving the proposed bound. The arXiv preprint is the only retrieved source supporting this development, so the disproof remains unverified.

Current status (as of August 2026): A preprint claims to disprove Magnanini’s conjecture by counterexample, but independent confirmation is not yet recorded.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Counterexamples to Magnanini's conjecture concerning critical points of the torsion function — arXiv — Edwards, Parker, Lundberg, Erik, Ramachandran, Koushik

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