Boundary comparison without global domination of linearized solutions

Establish linear boundary comparison for nonnegative solutions of the linearized fractional p-Laplacian without assuming a global domination condition of the form 0 ≤ F ≤ M_q U, including control of unbounded quotients and the exterior source term arising on the zero set of the background solution.

Background

Theorem 7.14 proves boundary comparison and Hölder regularity for the quotient F/U under the global bound 0 ≤ F ≤ M_q U. This assumption ensures that the quotient is bounded and that F vanishes on the entire zero set of U, thereby eliminating the source term in the transformed quotient equation.

The authors identify removing this assumption as an unresolved issue because an arbitrary nonnegative linearized solution may have an unbounded quotient and may generate a nonzero exterior source through interactions with the zero set of U. Resolving the problem would extend the linear comparison theory beyond the globally dominated class used in the deformation argument.

References

Two further questions concern the present operator. First, removing the global domination 0 ≤F ≤ MqU from linear boundary comparison requires control of unbounded quotients and the exterior source in (6.24). Second, a non-local counterpart of [26, Lemma 3.15] would give (9.9) C−1ε( ρ R)s+ε≤ u(Aρ(w))u(AR(w)) ≤ Cε( ρ R)s−ε, 0 < ρ < R. An iteration of flat comparisons might yield this sharper growth, with suitable control of the exterior data. A power bound of order s+ε is integrable at infinity in the natural (p−1)-tail when ε < s/(p−1). Neither question is settled or used in Theorem 1.1.

— Boundary Harnack inequalities for the fractional $p$-Laplacian in Reifenberg flat domains  (2610.21689 - Nyström, 15 Oct 2026) in Section 9.2, page 73

An NTA extension would still require the structural and local energy hypotheses of Lemma 7.11, together with normalization and maximum-lift bounds, uniformly over the ordered backgrounds. Under these inputs the quotient and deformation arguments need no approximating hyperplane. General NTA geometry alone has not been shown here to provide them.

— Boundary Harnack inequalities for the fractional $p$-Laplacian in Reifenberg flat domains  (2610.21689 - Nyström, 15 Oct 2026) in Section 9.1, page 72

A possible extension would impose quantitative conditions on the negative exterior data, for example smallness of Tail(u−, w, 4r)u(A) and Tail(v−, w, 4r)v(A) ,for solutions positive in the comparison patch. Whether such conditions suffice for a version of Theo- rem 1.1 requires additional boundary non-degeneracy and comparison arguments. No such extension is asserted here.

— Boundary Harnack inequalities for the fractional $p$-Laplacian in Reifenberg flat domains  (2610.21689 - Nyström, 15 Oct 2026) in Remark 1.7, page 3