2017/09/12 by Alessio Figalli, Joaquim Serra, Figalli, Alessio +1 · 5 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1709.04002
In the classical obstacle problem, the free boundary can be decomposed into\n"regular" and "singular" points. As shown by Caffarelli in his seminal papers\n citeC77,C98, regular points consist of smooth hypersurfaces, while singular\npoints are contained in a stratified union of C1 manifolds of varying\ndimension. In two dimensions, this C1 result has been improved to\nC1,\α by Weiss citeW99.\n In this paper we prove that, for n=2 singular points are locally contained\nin a C2 curve. In higher dimension n\≥ 3, we show that the same result\nholds with C1,1 manifolds (or with countably many C2 manifolds), up to\nthe presence of some "anomalous" points of higher codimension. In addition, we\nprove that the higher dimensional stratum is always contained in a\nC1,\α manifold, thus extending to every dimension the result in\n citeW99.\n We note that, in terms of density decay estimates for the contact set, our\nresult is optimal. In addition, for n\≥3 we construct examples of very\nsymmetric solutions exhibiting linear spaces of anomalous points, proving that\nour bound on their Hausdorff dimension is sharp.\n