For almost every obstacle, the degenerate part of the free boundary in the fractional obstacle problem vanishes up to dimension 3 for every s in (0,1).
An epiperimetric inequality for odd frequencies in the thin obstacle problem
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abstract
We prove for the first time an epiperimetric inequality for the thin obstacle Weiss' energy with odd frequencies and we apply it to solutions to the thin obstacle problem with general $C^{k,\gamma}$. In particular, we obtain the rate of convergence of the blow-up sequences at points of odd frequencies and the regularity of the strata of the corresponding contact set. We also recover the frequency gap for odd frequencies obtained by Savin and Yu.
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Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian
For almost every obstacle, the degenerate part of the free boundary in the fractional obstacle problem vanishes up to dimension 3 for every s in (0,1).