论文标题
薄障碍问题的自由边界的通用规律性
Generic regularity of free boundaries for the thin obstacle problem
论文作者
论文摘要
$ \ mathbb {r}^{n+1} $中的Signorini问题的自由边界在退化集外的外部平稳，该集合可以具有与自由边界本身相同的维度（$ n-1 $）。 在[fr21]中，一般来说，自由边界不光滑的集合最多是$（n-2）$ - 尺寸。我们的主要结果表明，实际上，退化集的零$ \ natercal {h}^{n-3-α_0} $用于通用解决方案。作为副产品，我们可以获得$ n+1 \ leq 4 $的副产品，整个自由边界通常是光滑的。这解决了Schaeffer在$ \ Mathbb {r}^3 $和$ \ Mathbb {r}^4 $中的猜想的类似物。
The free boundary for the Signorini problem in $\mathbb{R}^{n+1}$ is smooth outside of a degenerate set, which can have the same dimension ($n-1$) as the free boundary itself. In [FR21] it was shown that generically, the set where the free boundary is not smooth is at most $(n-2)$-dimensional. Our main result establishes that, in fact, the degenerate set has zero $\mathcal{H}^{n-3-α_0}$ measure for a generic solution. As a by-product, we obtain that, for $n+1 \leq 4$, the whole free boundary is generically smooth. This solves the analogue of a conjecture of Schaeffer in $\mathbb{R}^3$ and $\mathbb{R}^4$ for the thin obstacle problem.
