# How often does the gradient of a solution to elliptic equation vanish on the boundary?

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?