Generic regularity of free boundaries for the thin obstacle problem

Abstract

The free boundary for the Signorini problem in Rn+1 is smooth outside of a degenerate set, which can have the same dimension (n−1) as the free boundary itself.In [15] it was shown that generically, the set where the free boundary is not smooth is at most (n−2)-dimensional. Our main result establishes that, in fact, the degenerate set has zero Hn−3−α0 measure for a generic solution. As a by-product, we obtain that, for n+1≤4, the whole free boundary is generically smooth. This solves the analogue of a conjecture of Schaeffer in R3 and R4 for the thin obstacle problem.