Given a system of PDEs with $n_f$ functions $f_i(x_1, x_2 ... n_v)$ each of $n_v$ real variables how many equations $n_c$ does it take before the system is unsolvable in general? I am guessing that as long as $n_c \le n_f n_v$ then these systems are usually solvable and if $n_c = n_f n_v$ then this is usually a finite dimensional infinite set and beyond that they should not be expected to be solvable. (My question is flexible in that if letting the variables be complex leads to a conceptually simpler answer than you are welcome to make that assumption).
An example would be the cauchy-riemann equations where $u,v$ are our $n_f=2$ functions $x,y$ are our $n_v =2$ variables, and the $n_c=2$ equations are given as:
$$ \frac{\partial u }{\partial x} = \frac{\partial v}{\partial y} \\ \frac{\partial u }{\partial y} = -\frac{\partial v}{\partial x} $$
We know that this system is solvable and admits an extremely large number of solutions (namely the set of all analytic functions around a point $p$). But if we add a constraint like the differential equation $y' = y$ then the solution space becomes a finite dimensional infinite set, and (I think) adding any additional PDE constraints generally speaking makes it unsolvable.
The complex differential equation $y' = y$ is actually a system of two of PDEs given as
$$ \frac{\partial u}{\partial x} = u \\ \frac{\partial v}{\partial x} = v $$
And so it seems like the sentence "what is a complex analytic function $f$ such that $f' =f$?" is really a system of $n_c=4$ PDEs of $n_f=2$ functions and $n_v=2$ variables and since $n_c = n_f * n_v$ we expect a finite dimensional infinite set of solutions (Computing the exact dimension seems to be a hard problem).
Is my intuition on this matter correct?
I guess I am asking both
- How can I actually state my question formally (I'm using words like usually and generally and that isn't really rigorous)
- Does such a simple rule hold?