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.
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