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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 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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Something puzzles me. You seem to believe that $d_{\mathcal A}$ is not smooth near $\mathcal A$. I'm sure of the opposite. Singularities of $d_{\mathcal A}$ occur along the set $\mathcal C$ of curvature centers of $\mathcal A$ (the so-called 'cut locus' ?). Therefore the solution could be constructed by the method of characteristics between $\mathcal A$ and $\mathcal C$. It will be unique ans smooth. –  Denis Serre Feb 4 '11 at 8:01
Sorry: to clarify, I agree that $d_{\mathcal{A}}$ is smooth near $\partial \mathcal{A}$; I just didn't know the solution method to use here. You provided me with the solution, thank you! –  Eugene Feb 4 '11 at 15:39
To follow up: I'm able to construct a solution around $\partial \mathcal A$ by applying the method of characteristics within each chart, and (by compactness) patching a finite number of them together. However, this only works locally around the boundary. I have tried to extend the solution towards $\mathcal C$, as you suggested can be done. However, I'm running into difficulties. Is there a direct approach (possibly given in a book or paper you can cite) for extending the method of characteristics to give a solution all the way to $\mathcal C$? –  Eugene Feb 24 '11 at 4:55
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1 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.

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