Coercivity conjecture for nonlocal divergence-form operators

Let $K:\mathbb{R}^n\times\mathbb{R}^n\to[0,\infty)$ be a symmetric measurable kernel, $K(x,y)=K(y,x)$ , satisfying the hypotheses proposed in the cited coercivity conjecture. Determine whether the associated nonlocal divergence-form equation $\operatorname{PV}\int_{\mathbb{R}^n}(u(x)-u(y))K(x,y)\,dy=0$ in $B_2$ has a coercive weak formulation and whether every weak solution is locally Hölder continuous in $B_2$ . The supplied sources do not specify the full original hypotheses precisely; they report that the broad regularity assertion has a claimed counterexample, while coercivity and Hölder regularity are proved under an additional $L^p$ -integrability assumption on annuli with $p>1$ .

References

Primary source

arXiv

Additional references

Coercivity and regularity for nonlocal divergence-form operators with measurable kernels — arXiv — Anup Biswas

nonlocal-partial-differential-equations regularity-theory

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed progress

An unrefereed preprint claims a counterexample to the broad regularity picture and proves coercivity under stronger assumptions, but the original conjecture remains open.

The conjecture concerns coercivity and interior regularity for nonlocal divergence-form operators with merely measurable kernels. Earlier work established Hölder regularity under additional structural assumptions, but not in the full generality proposed.

Known results

Earlier work gives an affirmative Hölder-regularity result when the jump measures are absolutely continuous or supported on sufficiently many subspaces, under assumptions including uniform comparability to an $\alpha$ -stable measure, with $\alpha\in(0,2)$ ; the estimate holds for almost every pair of interior points.

October 2026 conditional theorem and counterexample

Anup Biswas's preprint claims a bounded discontinuous weak solution, ruling out the proposed broad regularity principle. It also claims coercivity and interior Hölder regularity under stronger integrability assumptions. These claims are reported from an unrefereed preprint and remain unverified.

Current status (as of October 2026): The broad conjecture remains open; a preprint claims a counterexample to the unrestricted regularity principle and a conditional coercivity and regularity theorem, but neither claim is independently verified.

Sources

No solutions have been posted yet.