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

Abstract

In this article we study a normalised double obstacle problem with polynomial obstacles $p^1(x)\le p^2(x)$, where the equality holds iff $x=0$. In dimension two we give a complete classification of blow-up solutions. In particular, we see that there exists a new type of blow-ups, which we call double-cone solutions, since the coincidence sets $\{u=p^1\}$ and $\{u=p^2\}$ are cones with a common vertex. Furthermore, we show that if the solution to the double obstacle problem has a double-cone blow-up limit at the origin, then the blow-up is unique, and locally the free boundary consists of four $C^{1,\gamma }$-curves, meeting at the origin.