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 ≤ 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 solutions depending on the coefficients of the polynomials p 1 , p 2 . In particular, we see that there exists a new type of blow-ups, that we call double-cone solutions since the coincidence sets {u = p 1 } and {u = p 2 } are cones with a common vertex.We prove the uniqueness of blow-up limits, and analyse the regularity of the free boundary in dimension two. In particular we show that if the solution to the double obstacle problem has a double-cone blow-up limit at the origin, then locally the free boundary consists of four C 1,γ -curves, meeting at the origin.In the end we give an example of a three-dimensional double-cone solution.