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We investigate the regularity of the free boundary for the Signorini problem in R n + 1 . It is known that regular points are ( n - 1 ) -dimensional and C ∞ . However, even for C ∞ obstacles φ , the set of non-regular (or degenerate) points could be very large-e.g. with infinite H n - 1 measure. The only two assumptions under which a nice structure result for degenerate points has been established are when φ is analytic, and when Δ φ 0 . Finally, we construct some new examples of free boundaries with degenerate points.We consider the Frö
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