# On the fine structure of the free boundary for the classical obstacle problem

## Abstract

In the classical obstacle problem, the free boundary can be decomposed into “regular” and “singular” points. As shown by Caffarelli in his seminal papers (Caffarelli in Acta Math 139:155–184, 1977; J Fourier Anal Appl 4:383–402, 1998), regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of \(C^1\) manifolds of varying dimension. In two dimensions, this \(C^1\) result has been improved to \(C^{1,\alpha }\) by Weiss (Invent Math 138:23–50, 1999). In this paper we prove that, for \(n=2\) singular points are locally contained in a \(C^2\) curve. In higher dimension \(n\ge 3\), we show that the same result holds with \(C^{1,1}\) manifolds (or with countably many \(C^2\) manifolds), up to the presence of some “anomalous” points of higher codimension. In addition, we prove that the higher dimensional stratum is always contained in a \(C^{1,\alpha }\) manifold, thus extending to every dimension the result in Weiss (1999). We note that, in terms of density decay estimates for the contact set, our result is optimal. In addition, for \(n\ge 3\) we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points, proving that our bound on their Hausdorff dimension is sharp.

## Notes

### Acknowledgements

both authors are supported by ERC Grant “Regularity and Stability in Partial Differential Equations (RSPDE)”. The authors thank the anonymous referees for useful comments on a preliminary version of the manuscript.

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