In this article we study a normalised double obstacle problem with polynomial obstacles $ 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 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 $C^{1,\gamma}$-curves, meeting at the origin. In the end we give an example of a three-dimensional double-cone solution.

中文翻译：

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二维双障碍问题的爆破分析

在本文中，我们在假设 $ p^1(x)=p^2(x)$ iff $ x=0$ 的情况下，研究了多项式障碍 $ p^1\leq p^2$ 的归一化双障碍问题。在维度二中，我们根据多项式 $p^1, p^2$ 的系数给出了爆破解决方案的完整特征。特别是，我们看到存在一种新的爆炸类型，我们称之为双锥解，因为巧合集 $\{u=p^1\}$ 和 $\{u=p^2\}$是具有公共顶点的锥体。我们证明了膨胀极限的唯一性，并分析了自由边界在二维上的规律性。特别地，我们表明，如果双障碍问题的解决方案在原点有一个双锥爆炸极限，那么局部自由边界由四个 $C^{1,\gamma}$ 曲线组成，在起源。