Mean curvature bounds for the obstacle problem

Giacomo Colombo, Federico Franceschini

arXiv:2608.19991v1 Primary: math.AP

Abstract

We prove optimal $(n-1)$-semiconvexity estimates for solutions to the classical obstacle problem, possibly with a smooth right-hand side. We deduce that the mean curvature of the free boundary is universally bounded above in all dimensions. Existing examples show that full curvature bounds fail in dimensions three and higher. Noticeably, these results are novel even for a constant right-hand side in the $n=2$ case.

Disclosure

“to the reader. At the same time, presenting only the general case from the start would likely obscure the main ideas. This motivates our two-step presentation. 1.5. Use of AI. Beside minor editing tasks, the authors acknowledge some use of AI models exclusively in the following: to speed up computations for the proof of Theorem 4.11, to prove the technical Theorem 2.7 and for adapting classical monotonicity formulas to the case with right hand side. The paper has entirely been written”
PDF page 5

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-22T03:11:21.810084Z

Structural counts

Pages 59 pdf

Theorems 4 source

Lemmas 35 source

Propositions 7 source

Corollaries 0 source

Definitions 0 source

Displayed equations 581 source

Bibliography entries 40 source

Appendix pages 59 estimated

Count notes

Source counts use the expanded primary TeX file paper_ND_RHS.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.