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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. We can assume that $\partial \Omega$ is connected.

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.

For non-constant boundary values $f$, at how many points can we have $\nabla u = 0$? In particular, $f$ might be constant on a large part of $\partial \Omega$.

Almost two-to-one boundary values

In two dimensions we can easily select boundary values so that the boundary can be divided into two parts; on one part, the boundary data $f$ is strictly increasing, and on the other, strictly decreasing. (Impose an orientation on the boundary first to make sense of this.) Then $\nabla u$ is non-zero everywhere except possibly at the maximum and minimum of $f$. For non-degenerate $p$-Laplace type equations it is known that $\nabla u$ is nonzero on all of $\partial \Omega$ in this case.

In higher dimension life is not quite so simple, but at least for strictly convex sets, using a linear function as $f$ gives much the same effect. I am sure this can be generalized much further.

But this type of boundary value is highly specific. What can be said for an arbitrary boundary value? Is there at most a finite or a countable set of points with zero gradient, for example?

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