We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded $1$-dimensional topological submanifold of the plane. Here monotonicity is in the sense of Lebesgue: the maximum and minimum of the function in an open set are attained at the boundary. Our result is an analog of Sard's theorem, which asserts that for a $C^2$-smooth function in a planar domain almost every value is a regular value. As an application we show that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions.

中文翻译：

---

平面域中的单调 Sobolev 函数：水平集和平滑逼近

我们证明了平面域中 Sobolev 函数的几乎每个水平集都由点、Jordan 曲线或区间的同胚副本组成。对于平面中的单调 Sobolev 函数，我们有更强有力的结论，即几乎每个水平集都是平面的嵌入 $1$ 维拓扑子流形。这里的单调性是 Lebesgue 意义上的：在边界处获得开集函数的最大值和最小值。我们的结果是 Sard 定理的模拟，它断言对于平面域中的 $C^2$-平滑函数，几乎每个值都是常规值。作为一个应用，我们展示了平面域中的单调 Sobolev 函数可以通过平滑单调函数在 Sobolev 范数中均匀近似。