A mathematical preprint has established a one-sided limit on the combined bending of the free boundary in a classical obstacle problem. Under a positive C1,α right-hand side, it proves that the boundary’s mean curvature—a measure of its combined bending—is bounded above locally, and that the function defining the boundary obeys a related semiconvexity bound.
The model uses a nonnegative function u that satisfies Δu=fχ{u>0} in the unit ball B1. Its free boundary is ∂{u>0}: the edge between the region where u is positive and the region where it is zero.
A bound that tightens near the boundary
At every point in B1/2 and for every chosen unit direction e, the paper bounds the sum of second derivatives in the directions perpendicular to e from below by minus a constant times the point’s distance from the free boundary. In ordinary terms, that transverse part of the solution cannot bend downward more sharply than a linear distance rule allows.
The constant depends only on the dimension and on the lower bound, regularity exponent and C1,α norm of f; the preprint does not supply a numerical value. The authors say the linear rate is optimal and cannot be improved even near a regular free-boundary point.
Mean curvature, tested from outside
The curvature statement is made in the viscosity sense, a way of testing a possibly nonsmooth boundary with smooth comparison surfaces. If an outer C2 surface E touches the free boundary while containing the local positive region, its mean curvature at the contact point is no greater than C0.
That is a one-sided, viscosity-based statement, not a two-sided classical curvature estimate at every free-boundary point.
A sharper two-dimensional case—and a higher-dimensional limit
In two dimensions with f≡1, the paper goes further: its full Hessian is bounded below by a linear distance term throughout B1/2. That matrix bound covers all second-derivative directions at once, rather than only the transverse combinations controlled in the general statement.
The stronger geometric control has a limit in higher dimensions. For n≥3, the document states that no universal upper bound on the full second fundamental form—the boundary’s curvature in all directions—is possible, citing a solution with f≡1 in B1⊂R3.
Another boundary on what can be guaranteed
A separate construction in Appendix C gives a smooth two-dimensional obstacle and solution whose contact set has points in B1/2 that cannot be touched from outside by a ball of any prescribed radius ρ>0. The paper therefore does not establish a universal exterior-ball radius.
A proof, not an experiment
This is a theoretical proof study, not an empirical investigation: it analyzes mathematical solutions of the obstacle problem and their free boundaries, with no participant group or reported empirical sample.
The proof combines frequency-formula arguments, improvement-of-semiconvexity methods, logarithmic epiperimetric inequalities, spectral gaps and linearization around integer-frequency points. It first treats a two-dimensional constant-right-hand-side model and then adapts the arguments to the general case.
The document is an arXiv preprint, identified as arXiv:2608.19991v1 and dated 20 Aug 2026.
Paper data and sources
Original title: Mean curvature bounds for the obstacle problem
Authors: Giacomo Colombo, Federico Franceschini
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text