If $u$ is a Sobolev function then $\nabla u = 0$ on $\{ u = c\}$.

Question

There is a result of the form:

If $u$ is a Sobolev function on some domain then $\nabla u = 0$ on $\{ x \mid u(x) = c\}$ where $c$ is constant.

Can someone point me to a specific reference? I cannot find it anywhere. I want to know the precise assumptions on the domain.

Answer 1

On Evans and Gariepy's book, page 130, Theorem 4 (iv), is the result you want. Remember "a.e." is the key element here.

Answer 2

The result is:

If $u \in W^{1,p}(\Omega)$ on an open set $\Omega$, and if $A \subset \mathbb{R}$ is a null set, then $\nabla u = 0$ a.e. on $u^{-1}(A)$.

The result is due to Stampacchia. If anyone can refer me to an English proof I'd be grateful.

My source is Nonlinear Analysis by Gasinski and Papageorgiou (Remark 2.4.26).