A dichotomy for everywhere differentiable eikonal functions

Question

Let $f: \mathbb R^n \to \mathbb R$ be everywhere differentiable, with $|\nabla f| = 1$ almost everywhere. Is it true that $|\nabla f| = 0$ or $1$ everywhere?

Answer 1

Apply Theorem 4.2 in this paper (which I learned from your comment) with $U = B_1(- e_1/2)$, and obtain a $\mathbb{Z}^n$-periodic function $u$ on $\mathbb{R}^n$ such that $u$ is everywhere differentiable, $\nabla u(x)$ vanishes for $x$ in the boundary of the unit cube, $|\nabla u|\in\bar{U}$ everywhere in $\mathbb{R}^n$ and $\nabla u \in \partial U$ almost everywhere in $\mathbb{R}^n$. In particular, $|\nabla u(x) + \frac12 e_1 | =1$ for almost every $x\in\mathbb{R}^d$. Now let us define $f(x) := \frac12 e_1 \cdot x + u(x)$. Observe that $\nabla f = \frac12 e_1$ on the boundary of the unit cube and $|\nabla f|=1$ almost everywhere in $\mathbb{R}^n$.

Answer 2

Your equation $| \nabla f| = 1$ is a well-studied PDE called the eikonal equation in physics which in turn is perhaps the simplest version of the Hamilton–Jacobi equation. Suppose $f$ is continuous so that $C = f^{-1}(0)$ is closed. Then $f(x) =\operatorname{dist}(x, C)$ is the distance of $x$ to $C$ provided $f(x) > 0$ and $f$ is smooth at $x$. This can be seen by observing that the gradient flow of $f$ is given by $\phi_t (x_0) = x_0 + t v$ where $v =\nabla f (x_0)$ provided we are in a small enough neighborhood of a smooth point $x_0$ of $f$. Next observe that $f(\phi_t(x_0)) = f(x_0) + t$. This process can be reversed. Start with the closed set $C$ and form $f(x) =\operatorname{dist}(x, C)$. Then $f$ solves the eikonal equation where it is smooth. For example, take $C$ to be two points $A, B$ in the plane. Then $f$ fails to be differentiable exactly along the perpendicular bisector of the line segment $AB$. Thinking about these examples we see that it is not a good question really, to insist that $\|\nabla f \|$ be zero where it is not $1$. (In any case $f$ is certainly not $C^1$ if this happens!) Rather, such an $f$ is Lipschitz everywhere and fails to be $C^1$ along an interesting closed set, the “cut locus” of $C$. You can see some of this done in the following paper:

Albano, P.; Cannarsa, P.; Nguyen, Khai T.; Sinestrari, C., Singular gradient flow of the distance function and homotopy equivalence, Math. Ann. 356, No. 1, 23-43 (2013), MR3038120, Zbl 1270.35012.