This question is motivated by an inverse coefficient problem, for which it is useful to find solutions to a particular PDE so that the gradient of the solution does not vanish at all, or at least too often.

Suppose $\Omega \subset \mathbb{R}^d$ with $d \ge 2$ is a bounded open set with at least $C^1$ boundary. Suppose $f \colon \partial \Omega \to \mathbb{R}$ is at least $C^1$ smooth.

Consider the $p$-conductivity equation $$ \nabla \cdot (\sigma |\nabla u|^{p-2} \nabla u) = 0 $$ with the Dirichlet boundary conditions $f$. Suppose that $\Omega$, $f$ and $\sigma$ are smooth enough so that we have $u \in C^1(\overline{\Omega})$. Here $\sigma \in L^\infty(\Omega)$ is bounded from below by a positive constant; $\sigma$ is called the conductivity. We suppose it is continuous, at least near the boundary.

**When can we say $\nabla u = 0$ only rarely on $\partial \Omega$? What conditions must $f$ satisfy?** To be more precise, I would like the set where $\nabla u \neq 0$ (intersected with $\partial \Omega$) to be dense in $\partial \Omega$.

My best guess is that if I select nonconstant boundary values $f$, then $\nabla u$ should vanish only rarely. If $p = 2$ and everything is smooth, then unique continuation over a hypersurface suggests this.

Another approach would be to consider $\nabla f$ restricted to $\partial \Omega$ and argue that, for general (generic?) boundary values $f$, this gradient rarely vanishes. Is there a way to state something like this formally, and is this even true?

The weakest relevant statement would be to prove that there exist boundary values with the desired property. This seems to be true, at least.