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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 $2N_y=0$. We get $N(x,y)=F(x)=z$ 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

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Uri: for the sake of my eyes, can you please try to use the LaTeX capabilities of this site to make your expressions more readable? (Look to your right and read the "How to write math" box.) –  Willie Wong Aug 3 '11 at 21:19
3  
Yes, the problem is exactly that the plane $z=0$ is a characteristic surface for this (overdetermined) system of PDE. In fact, the surfaces on which $dz$ vanishes are precisely the domain surfaces such that no amount of information about $G$ and its derivatives along the surface will be sufficient to determine a solution $G$ in a neighborhood of the surface. Overdetermined systems don't generally break up into hyperbolic', elliptic', or `parabolic' in any natural way without knowing more about the system. This system does have real, distinct characteristics, though. –  Robert Bryant Aug 3 '11 at 22:05
1  
Uri, I've edited the latex, please check that I haven't changed the meaning of what you originally wrote. –  David Roberts Aug 4 '11 at 4:48
    
I'm not sure what more to say. Yes, adding $G_{xx}-G_{zz}=0$ to the other $4$ equations yields a system with empty characteristic variety, so the solutions of the $5$-equation system depend only on constants. (Note that adding $G_{xx}=0$ instead would not eliminate the characteristics.) I guess that you would like to know how to compute such things for more general systems, i.e., how to determine the generality of solutions and how Cauchy data are properly posed for overdetermined systems of PDE. There is a theory; for example, see Exterior Differential Systems (our book). –  Robert Bryant Aug 4 '11 at 11:28
1  
as you seem to want a tutorial, try posting at math.stackexchange.com/questions?sort=active –  Will Jagy Aug 4 '11 at 21:57

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