Half-Space Solutions with 7/2 Frequency in the Thin Obstacle Problem

For the thin obstacle problem in $${\mathbb {R}}^3$$ , we show that half-space solutions form an isolated family space of $$\frac{7}{2}$$ -homogeneous solutions. a general solution with one blow-up profile this family, establish rate convergence to profile. As consequence, obtain regularity free boundary near such contact points.