Question

Hello, I have the following 4 PDEs which I am trying to solve for $G(x,y,z)$ :

(1) $G_{xy}=0$

(2) $G_{xz}=0$

(3) $G_{yz}=0$

(4) $G_{xx}-G_{yy}=0$.

It is not hard to see that the general solution is:

$$G(x,y,z)=C_1+C_2x+C_3y+C_4(x^2+y^2)+F_1(z).$$

Here $C$ represents constants and $F_1$ a function.

I am trying to approach this solution as follows: Do $(1)_{x}$-$(4)_{y}$ to get $G_{yyy}(x,y,z)=0$. Now consider this along the $y$-axis, i.e. $G_{yyy}(0,y,0)=0$. This in a third order ODE so its solution $G(0,y,0)$ involves 3 constants. Next integrate (1) w.r.t $y$ and get an arbitrary function of $x$ and $z$. Consider this function along the $y$-axis $(x=z=0)$ and it becomes an arbitrary constant i.e. $G_x(0,y,0)$ is a constant. So we end up with Cauchy data along the $y$ axis in terms of 4 constants. This Cauchy data together with equation (4) allows us to solve for $G(x,y,0)$, that is, to get the solution on the $x$-$y$ plane.

My next step was to extend this to the $z$ axis to get $G(x,y,z)$. Now for Cauchy data I need $G(x,y,0)$ which I already have, and the normal derivative $G(x,y,0)_z$ . Using equations (2) and (3) we can get $G(x,y,0)_z$ in terms of one additional constant, to a total of 5 constants. Now (finally...) to my question!

Given $G(x,y,0)$ and $G(x,y,0)_z$ I have Cauchy data in the $x$-$y$ plane that can be used to solve a PDE for $G(x,y,z)$, but I guess non of equations (1)..(4) or their combinations are that (hyperbolic) PDE since the general solution contains an arbitrary function $F_1(z)$. Are equations (1)..(4) parabolic? is the problem that the Cauchy data is on a charateristic surface?? I need a little help explaining this. Finally, please note that if I add a fifth equation (5) $G_{xx}-G_{zz}=0$ we get:

$$G(x,y,z)=C_1+C_2x+C_3y+C_4z+C_5(x^2+y^2+z^2)$$

that is, no arbitrary functions, just constants.

Is it trivial to conclude that $z=0$ is a charateristic plane to equations (1)-(4) but not to the fifth one? Here is a procedure I found which does not seem so trivial:

(a) Consider a charateristic surface $P(x,y,z)$. we have:$A_{ij}P_iP_j=0$ where $A$ is a 3X3 matrix containing the coefficients of the second order derivatives. The subscripts of$P$ denote differentiation and the summation is implied.

(b) Consider eq. $G_{yz}=0$ and a characteristic surface $z=N(x,y)$. Using (a) we get $A_{23}=A_{32}=1$ yielding $2N_x=0$2N_y=0$. We get $N(x,y)=F(y)=z$ N(x,y)=F(x)=z$ with $F$ arbitrary function. Take $F=0$ and get $z=0$ char surface.

(c) $G_{xz}$ gives $z=0$ char surface the same way.

(d) For $G_{xy}$ I got $F(z)=x$ with $F$ arbitrary. Choosing it large can take us as close as we want to $z=0$, e.g. take $F(z)=10^{10}z$. So, again $z=0$ is a charactristic.

(e) Similar arguments apply for $G_{xx}-G_{yy}$

(f) Finally regarding the fifth equation: Using this procedure, the equation $G_{xx}-G_{zz}$ does not give $z=0$ as characteristic surface, and therefore we can use the Cauchy data there to generate the full space solution.

Does this make sense or is this over complicating things?

Thanks

Uri