We investigate the regularity of the free boundary for the Signorini problem in \({\mathbb {R}}^{n+1}\). It is known that regular points are \((n-1)\)-dimensional and \(C^\infty \). However, even for \(C^\infty \) obstacles \(\varphi \), the set of non-regular (or degenerate) points could be very large—e.g. with infinite \({\mathcal {H}}^{n-1}\) measure. The only two assumptions under which a nice structure result for degenerate points has been established are when \(\varphi \) is analytic, and when \(\Delta \varphi < 0\). However, even in these cases, the set of degenerate points is in general \((n-1)\)-dimensional—as large as the set of regular points. In this work, we show for the first time that, “usually”, the set of degenerate points is small. Namely, we prove that, given any \(C^\infty \) obstacle, for almost every solution the non-regular part of the free boundary is at most \((n-2)\)-dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian \((-\Delta )^s\), and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is \((n-1-\alpha _\circ )\)-dimensional for almost all times t, for some \(\alpha _\circ > 0\). Finally, we construct some new examples of free boundaries with degenerate points.

我们研究\（{\ mathbb {R}} ^ {n + 1} \）中Signorini问题的自由边界的规律。已知正则点是\（（n-1）\）维和\（C ^ \ infty \）。但是，即使对于\（C ^ \ infty \）障碍物\（\ varphi \），非规则（或简并）点的集合也可能非常大，例如具有无限的\（{\ mathcal {H}} ^ { n-1} \）测量。建立退化点的良好结构结果的唯一两个假设是\（\ varphi \）解析时和\（\ Delta \ varphi <0 \）时。但是，即使在这些情况下，简并点的集合通常也是\（（n-1）\）维度-与常规点集一样大。在这项工作中，我们第一次证明“退化”点的集合通常很小。即，我们证明，给定任何\（C ^ \ infty \）障碍，几乎对于每个解决方案，自由边界的不规则部分最多为\（（n-2）\）维。这是Signorini问题朝这个方向的第一个结果。此外，我们证明了分数Laplacian \（（-\ Delta）^ s \）的障碍问题和抛物线型Signorini问题的相似结果。在抛物线型Signorini问题中，我们的主要结果表明，自由边界的非规则部分为\（（n-1- \ alpha _ \ circ）\）几乎所有时间t都是三维的，对于某些\（\ alpha _ \ circ> 0 \）。最后，我们构造了一些带有退化点的自由边界的新例子。