Good morning everybody. My question is inspired from the following fact:


Consider $\mathbb R^3$ endowed with coordinates $(x,y,z)$. Of course if we were to solve the second order pde $\partial\_x^2 g(x,y,z)=0$ we would find $g(x,y,z)=x p\_0(y,z)+p\_1(y,z)$. Then it will be possible, playing with $p\_0$ and $p\_1$, to prescribe for example both the behaviour on the hyperplanes $x=0$ and $x=1$ (Am I right?)


The question then is as follows: Suppose $X,Y,Z$ to be three arbitrary linearly independent vector fields (still in $\mathbb R^3$ with standard coordinates) and assume we want to solve the equation $X^2 g(x,y,z)=0$.


Assume $\pi\_0$ and $\pi\_1$ to be two hypersurfaces transverse to $X$ (they play the same roles of the previous hyperplanes).


Is it possible as above to find $g$ that satisfies ALSO prescribed conditions both on $\pi\_0$ and $\pi\_1$?


Intuitively this seems to be right to me, however I am not in possess of a rigorous proof, nor I know enough general theory to convince myself (therefore references are welcomed).


Thanks in advance for your kindness.


Regards,


-Guido-


**Edit** In response to prof. Bryant comment, let's say that my case of interest concerns with $X=\partial\_x-\frac y2\partial\_z$, but still I have no particular constraints on $\pi\_0$ and $\pi\_1$, except for them being different and both transverse to $X$.



Boundary behaviour of a second order pde with characteristics
Let us look at the local problem: taking $X$ a non-zero smooth vector field in a neighborhood of 0 in $\mathbb R^3$, you may choose local coordinates such that $X=\partial\_z$. If $π\_1, π\_2$ are smooth hypersurfaces such that $X$ is transverse to both of them, you may assume that they are given locally by
$$
π\_j=\{(x,y,z), z=f\_j(x,y)\}, j=1,2.
$$
The matter is now to solve
$$
\partial\_z^2u=0,\quad u(x,y,f\_j(x,y))=w\_j(x,y),
$$
i.e.
$
u(x,y,z)=g\_0(x,y) +zg\_1(x,y),$
$$
g\_0(x,y) +f\_1(x,y)g\_1(x,y)=w\_1(x,y),\quad
g\_0(x,y) +f\_2(x,y)g\_1(x,y)=w\_2(x,y).
$$
Since the $f\_j, w\_j$ are given, it means that we want to solve a $2\times 2$ linear system with unknowns $g\_0,g\_1$: a simple sufficient condition for this to be possible is that
$$
f\_1-f\_2\not=0,
$$
which is true geometrically when $π\_0, π\_1$ are disjoint transverse hypersurfaces to whom $X $ is transverse.
I guess that the global problem could be more complicated.


