Let n≥ 2, let u solve partial_t^2u-Delta_xu=0 on R^n+1 with u(·,0)=u_0∈ H^s(R^n) and partial_tu(·,0)=u_1∈ H^s-1(R^n), where s>tfrac12. Define the divergence set E to be the set of x∈R^n at which u(x,t)not→ u_0(x) or partial_tu(x,t)not→ u_1(x) as t→0. The Barceló–Bennett–Carbery–Rogers conjecture predicts the sharp Hausdorff-dimension bound dim_ HE≤ n+2-4s in the low-regularity range where this quantity is relevant.

Barceló–Bennett–Carbery–Rogers pointwise-convergence conjecture

Let $n\geq 2$ , let $u$ solve $\partial_t^2u-\Delta_xu=0$ on $\mathbb{R}^{n+1}$ with $u(\cdot,0)=u_0\in H^s(\mathbb{R}^n)$ and $\partial_tu(\cdot,0)=u_1\in H^{s-1}(\mathbb{R}^n)$ , where $s>\tfrac12$ . Define the divergence set $E$ to be the set of $x\in\mathbb{R}^n$ at which $u(x,t)\not\to u_0(x)$ or $\partial_tu(x,t)\not\to u_1(x)$ as $t\to0$ . The Barceló–Bennett–Carbery–Rogers conjecture predicts the sharp Hausdorff-dimension bound $\dim_{\mathrm H}E\leq n+2-4s$ in the low-regularity range where this quantity is relevant.

References

Primary source

arXiv

Additional references

The divergence set for the wave equation in higher dimensions — arXiv — Xiumin Du, Terence L. J. Harris, Jianhui Li

harmonic-analysis pde

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A 2026 paper improves the answer for wave equations in four dimensions, but the full conjecture remains open beyond the settled low-dimensional cases.

The Barceló–Bennett–Carbery–Rogers conjecture predicts the sharp Hausdorff dimension of divergence sets for the wave equation at low Sobolev regularity. It is known in dimensions $d=2$ and $d=3$ , while the higher-dimensional conjecture remains unresolved.

Known results

Barceló et al.: the $d=2$ case was determined sharply.
Ham, Ko, and Lee (2021): the conjecture was proved for $d=3$ ; for $d\ge4$ they obtained improved upper bounds, not the full conjecture.

September 2026 partial advance

Xiumin Du, Terence L. J. Harris, and Jianhui Li report pointwise convergence outside a Hausdorff-dimension exceptional set for the four-dimensional wave equation when $0.5<s\le0.55$ , together with a broader partial bound. This advances but does not settle the conjecture in $d=4$ or higher; the result is reported here as unverified.

Current status (as of September 2026): The conjecture is settled for $d=2$ and $d=3$ , and partially advanced for $d=4$ in $0.5<s\le0.55$ ; the full higher-dimensional conjecture remains open.

Sources

No solutions have been posted yet.