Analysis of blow-ups for the double obstacle problem in dimension two

In this article we study a normalized double obstacle problem with polynomial obstacles p ≤ p under the assumption p(x) = p(x) iff x = 0. In dimension n = 2 we give a complete characterization of blow-up solutions depending on the coefficients of the polynomials p, p. We see that there exists a new type of blow-ups, that we call double-cone solutions since the noncoincidence set is a union of two cones with a common vertex. The main object of investigation is the double obstacle problem, having rotational invariant double-cone blow-up solutions, which happens if p(x) = −x1 −x2 and p(x) = x1 +x2. Assuming that the solution is close to a double-cone solution in the unit ball B1 ⊂ R, we prove that in a neighborhood of the origin the free boundary is a union of four C-graphs, pairwise crossing at the origin.