The eikonal equation in geometric equation is
$$ n(x, y, z) = \left( \frac{\partial S}{\partial x } \right)^2+\left( \frac{\partial S}{\partial y } \right)^2 +\left( \frac{\partial S}{\partial z } \right)^2 . $$
Here $n $ is a real function. For an arbitrary surface $C$, if we take it as a constant-value surface for $S$, we get a boundary value problem and we can try to determine $S $ for the whole 3-dimensional space.
The problem is, is this always possible? If not, for what kind of surface it is possible? For simplicity, first let us consider the case $n \equiv 1 $.
