On the fine structure of the free boundary for the classical obstacle problem
By Alessio Figalli, J. Serra. Inventiones Mathematicae, 2017
83 citations recorded by the source corpus.
AbstractIn 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 $$C^1$$C1 manifolds of varying dimension. In two dimensions, this $$C^1$$C1 result has been improved to $$C^{1,\alpha }$$C1,α by Weiss (Invent Math 138:23–50, 1999). In this paper we prove that, for $$n=2$$n=2 singular points are locally contained in a $$C^2$$C2 curve. In higher dimension $$n\ge 3$$n≥3, we show that the same result holds with $$C^{1,1}$$C1,1 manifolds (or with countably many $$C^2$$C2 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 $$C^{1,\alpha }$$C1,α 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 $$n\ge 3$$n≥3 we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points, proving that our bound on their Hausdorff dimension is sharp.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/75c3425a706701ad51a4910552924b4b4c710dc1
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In Applied Mathematics
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- Maria Colombo
- Bozhidar Velichkov
- Patrick Winkert
- Giuseppe Mingione
- Leszek Gasinski
- Cristiana De Filippis
- P. Hasto
In Geometry and Topology
Citations in the corpus
- A logarithmic epiperimetric inequality for the obstacle problem (2017, background): “3) with γ = 0) cannot hold in a neighborhood of the set S; this follows from the counterexample to the C1,α regularity of the singular free boundaries given in [11].”