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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 can 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, $\partial \mathcal{A}$?

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# Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

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 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, $d_{\mathcal{A}}$ 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 can 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, $\partial \mathcal{A}$?