Sobolev functions on $\mathbb{R}^N$ cannot be discontinuous on a $(N-1)$-dimensional submanifold

Question

How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?

Answer 1

Any function in $W^{1,p}$, $p>N$, has a continuous representative by the Sobolev embedding theorem so there is no issue here. However:

Proposition. There is a function $f\in W^{1,p}$, $p\leq N$, that is essentially discontinuous everywhere. In fact you can find a function such that the essential supremum on every open set is $+\infty$ and the essential infimum on every open set is $-\infty$.

See Example 2.26 in [2]. For a similar construction see also:

A function in $W^{1,p}(\Omega)$ for $1 < p < n$ which is not differentiable a.e.

Thus the answer to the question the way it is stated is no, the function can be essentially discontinuous everywhere.

There is however, a different point of view which shows that, in fact, a Sobolev function behaves nicely when restricted to an $(N-1)$-dimensional manifold and I will present two different approaches to it.

Approach 1. According to Theorem 2 p. 164 in [1] (I am referring to the first edition) any function $f\in W^{1,p}$ has a representative that is absolutely continuous on almost all lines. Here by a representative I mean a Borel function defined everywhere and equal to $f$ almost everywhere.

If $M\subset\Omega$ is an $(N-1)$-dimensional manifold, then almost all lines pass through almost all points on $M$ so we can define restriction of $f$ to $M$ by looking at values of $f$ at the points where the lines intersect with $M$. Such a restriction is called a trace. Therefore $f$ may be discontinuous on $M$, but still it behaves nicely on $M$.

Approach 2. Any Function in $W^{1,p}$, $p>N$, has a continuous representative by the Sobolev embedding theorem so there is no issue here.If $f\in W^{1,N}$, then $f\in W^{1,p}_{\rm loc}$ for ant $1\leq p<N$ so it suffices to discuss the case $1\leq p<N$ only.

The following result is Theorem 1 on p. 160 in [1].

Theorem 1. There is a representative of $f\in W^{1,p}(\Omega)$, $1\leq p<N$, $\Omega\subset\mathbb{R}^N$ that is $p$-quasicontinuous. That means for any $\epsilon>0$, there is an open set $V\subset\Omega$ with $\operatorname{Cap}_p(V)<\epsilon$ such that $f|_{\Omega\setminus V}$ is continuous.

Here $\operatorname{Cap}_p$ stands for the $p$-capacity.

Capacity is a certain outer measure. While I will not recall its definition I will explain how it is related to the Hausdorff measure. The next result is Theorem 4 p. 156 and and Theorem 3 p. 193 in [1].

Theorem 2. If $1\leq p<N$ and $\operatorname{Cap}_p(A)=0$, then $\mathcal{H}^s(A)=0$ for all $s>n-p$. Moreover if $A$ is compact, then $\operatorname{Cap}_1(A)=0$ if and only if $\mathcal{H}^{N-1}(A)=0$.

Now Theorem 1 says that away of a set of arbitrarily small capacity, $f$ is continuous so the exceptional set has capacity zero and Theorem 2 says that this exceptional set has vanishing $(N-1)$-dimensional measure. Therefore is we consider an $(N-1)$-dimensional manifold $M$ in $\Omega$ this exceptional set has measure zero.

[1] L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.

[2] P. Hajłasz, Non-linear elliptic partial differential equations .