For every integer n≥ 3, there exists a constant C_n>0 such that, for every nonzero real-valued function u satisfying Delta u=0 in B_4⊂R^n and u(0)=0, one has H^n-1bigl(x∈ B_2:u(x)=0bigr)≥ C_nN(u), where N(u)=log_2!(fracsup_B_1|u|sup_B_1/2|u|). The dependence on N(u) is conjectured to be optimal.

Folklore conjecture on nodal volume of harmonic functions

For every integer $n\ge 3$ , there exists a constant $C_n>0$ such that, for every nonzero real-valued function $u$ satisfying $\Delta u=0$ in $B_4\subset\mathbb{R}^n$ and $u(0)=0$ , one has $\mathcal{H}^{n-1}\bigl(\{x\in B_2:u(x)=0\}\bigr)\ge C_n\mathcal{N}(u)$ , where $\mathcal{N}(u)=\log_2\!\left(\frac{\sup_{B_1}|u|}{\sup_{B_{1/2}}|u|}\right)$ . The dependence on $\mathcal{N}(u)$ is conjectured to be optimal.

References

Primary source

arXiv

Additional references

A sharp lower bound for the nodal volume of harmonic functions — arXiv — Zhehui Wang

geometric-measure-theory harmonic-analysis partial-differential-equations

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to establish the conjectured lower bound, but the result has not yet been independently verified.

The conjecture predicts a sharp dimension-dependent lower bound for the nodal hypersurface measure of a harmonic function in a Euclidean ball. The target is the linear dependence on the function's oscillation or frequency quantity.

Known results

In March 2023, an almost-sharp result proved a bound with exponent $1-\varepsilon$ for $n\ge 3$ , while explicitly leaving the linear estimate as Conjecture 1.3.

September 10, 2026 claimed proof

Zhehui Wang's preprint A sharp lower bound for the nodal volume of harmonic functions claims the conjectured dimension-dependent bound in the stated Euclidean-ball setting. The proof is available only as an unrefereed preprint, so this is a claimed resolution rather than a verified theorem.

Current status (as of September 2026): The conjecture has a claimed proof in an unrefereed preprint, but independent verification is not recorded.

Sources

No solutions have been posted yet.