We consider the singular set in the thin obstacle problem with weight $|x_{n+1}{|}^a$ for $a\in \left(-1,1\right)$, which arises as the local extension of the obstacle problem for the fractional Laplacian (a nonlocal problem). We develop a refined expansion of the solution around its singular points by building on the ideas introduced by Figalli and Serra to study the fine properties of the singular set in the classical obstacle problem. As a result, under a superharmonicity condition on the obstacle, we prove that each stratum of the singular set is locally contained in a single $C^2$ manifold, up to a lower-dimensional subset, and the top stratum is locally contained in a $C^{1,\alpha }$ manifold for some $\alpha >0$ if $a<0$.

In studying the top stratum, we discover a dichotomy, until now unseen, in this problem (or, equivalently, the fractional obstacle problem). We find that second blow-ups at singular points in the top stratum are global, homogeneous solutions to a codimension-2 lower-dimensional obstacle problem (or fractional thin obstacle problem) when $a<0$, whereas second blow-ups at singular points in the top stratum are global, homogeneous, and $a$-harmonic polynomials when $a\ge 0$. To do so, we establish regularity results for this codimension-2 problem, which we call the very thin obstacle problem.

Our methods extend to the majority of the singular set even when no sign assumption on the Laplacian of the obstacle is made. In this general case, we are able to prove that the singular set can be covered by countably many $C^2$ manifolds, up to a lower-dimensional subset.

我们考虑带权重的薄障碍问题中的奇异集 $|{\times }_{n+1}{|}^{\mathrm{一个}}$ 对于 $\mathrm{一个}\in \left(-1,1\right)$，它作为分数拉普拉斯算子（非局部问题）的障碍问题的局部扩展而出现。我们在 Figalli 和 Serra 引入的思想的基础上，围绕其奇异点开发了解决方案的精细扩展，以研究经典障碍问题中奇异集的优良特性。结果，在障碍物的超调和条件下，我们证明奇异集合的每个层都局部包含在单个${丙}^2$ 流形，直到一个较低维的子集，并且顶层局部包含在一个 ${丙}^{1,\alpha }$ 对某些人来说是多方面的 $\alpha >0$ 如果 $\mathrm{一个}<0$.

在研究顶层时，我们发现了这个问题（或等价的分数障碍问题）中的二分法，直到现在还没有发现。我们发现，当$\mathrm{一个}<0$，而顶层奇异点的第二次爆炸是全局的、同质的，并且 $\mathrm{一个}$- 谐波多项式当 $\mathrm{一个}\ge 0$. 为此，我们为这个 codimension-2 问题建立了规律性结果，我们称之为非常薄的障碍问题。

即使没有对障碍物的拉普拉斯算子进行符号假设，我们的方法也可以扩展到奇异集的大部分。在这个一般情况下，我们能够证明奇异集可以被可数多${丙}^2$ 流形，直到一个低维子集。