Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, uniqueness, and smoothness of the solution $Z$ to the following PDE:
$$
Z(x) \Delta d_{\mathcal{A}}(x) + 2 \nabla d_{\mathcal{A}}(x) \cdot \nabla Z(x) = 0
$$
for $x \in \mathcal{M} \backslash \mathcal{A}$. Here $d_{\mathcal{A}}$ is the distance from point $x$ to $\mathcal{A}$. The Dirichlet boundary conditions are
$$
Z(x) = W(x)
$$
for $x \in \partial \mathcal{A}$, where $W \in C^{\infty}(\mathcal{A})$.
In general, $\nabla d_{\mathcal{A}}$ is locally of special bounded variation, and absolutely continuous for small enough open balls around $\partial \mathcal{A}$.
The PDE above does not admit a global continuous solution. None of the methods for viscosity solutions of Hamilton-Jacobi problems apply even for open sets $U$ near $\partial \mathcal{A}$. Can anything be said about the existence, uniqueness, continuity, and possibly more general smoothness of $Z$ for open balls $U$ within the singularity set (the generalized "cut locus"), i.e. near $\partial \mathcal{A}$?
Best Answer
Let $\Phi_t(x)$ be the flow of your vector field $\nabla d_{\mathcal{A}}$, starting at $x \in \mathcal{A}$. It will exist until the time $t_0(x)$, when it hits the set where $\nabla d_{\mathcal{A}}$ is not smooth.
Consider the ODE $$2 \frac{\partial}{\partial t} z(t, x) + z(t, x)\cdot \Delta d_{\mathcal{A}}(\Phi_t(x)) = 0, ~~~~~~~ z(0, x) = W(x)$$ depending on the parameter $x \in \partial \mathcal{A}$. It is a linear ODE, hence it has a unique solution that exists for all time, or rather until $t_0(x)$ (because afterwards the equation does not make sense anymore). It is also well-known that the solution depends smoothly on the parameter $x$, as all the data is smooth.
Now let $\mathcal{B} = \{\Phi_t(x) \mid x \in \partial \mathcal{A}, t < t_0(x)\} \subseteq \mathcal{A}$ be the set of points $y$ that lie on a unique flowline, i.e. there exist a unique $t$ and a unique $x \in \partial \mathcal{A}$ such that $y = \Phi_t(x)$ (that this set has the claimed description is because flowlines cannot cross). This is a neighborhood of $\partial \mathcal{A}$ in $\mathcal{A}$, and its complement in $\mathcal{A}$ has measure zero.
For points in $\mathcal{B}$, set $$Z(\Phi_t(x)) := z(t, x).$$
This gives you a solution that has the desired boundary values and is smooth on $\mathcal{B}$.
However, you know nothing about the smoothness in $\mathcal{A} \setminus \mathcal{B}$. For example, if you take a disc in $\mathbb{R}^n$, your function $Z$ will be continuous if and only if $W(x) \equiv \mathrm{const}$. Otherwise it will not be continuous at the origin.