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In this article we study a normalised double obstacle problem with polynomialobstacles $ p^1\leq p^2$ under the assumption that $ p^1(x)=p^2(x)$ iff $ x=0$.In dimension two we give a complete characterisation of blow-up solutionsdepending on the coefficients of the polynomials $p^1, p^2$. In particular, wesee that there exists a new type of blow-ups, that we call double-conesolutions since the coincidence sets $\{u=p^1\}$ and $\{u=p^2\}$ are cones witha common vertex. We prove the uniqueness of blow-up limits, and analyse theregularity of the free boundary in dimension two. In particular we show that ifthe solution to the double obstacle problem has a double-cone blow-up limit atthe origin, then locally the free boundary consists of four$C^{1,\gamma}$-curves, meeting at the origin. In the end we give an example ofa three-dimensional double-cone solution.

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