PDE’s and Frobenius Theorem

I’ve started many blog posts on algebra/algebraic geometry, but they won’t get finished and posted for a little while. I’ve been studying for a test I have to take in a few weeks in differential geometry-esque things. So I’ll do a few posts on things that I think are usually considered pretty easy and obvious to most people, but are just things I never sat down and figured out. Hopefully this set of posts will help others who are confused as I recently was.

My first topic is about the Frobenius Theorem. I’ve posted about it before. Here’s the general idea of it: If {M} is a smooth manifold and {D} is a smooth distribution on it, then {D} is involutive if and only if it is completely integrable (i.e. there is are local flat charts for the distribution).

What does this have to do with being able to solve partial differential equations? I’ve always heard that it does, but other than the symbol {\displaystyle\frac{\partial}{\partial x}} appearing in the defining of a distribution or of the flat chart, I’ve never figured it out.

Let’s go through this with some examples. Are there any non-constant solutions {f\in C^\infty (\mathbb{R}^3)} to the systems of equations: {\displaystyle \frac{\partial f}{\partial x}-y\frac{\partial f}{\partial z}=0} and {\displaystyle \frac{\partial f}{\partial y}+x\frac{\partial f}{\partial z}=0}?

Until a few days ago, I would have never thought we could use the Frobenius Theorem to do this. Suppose {f} were such a solution. Define the vector fields {\displaystyle X=\frac{\partial}{\partial x}-y\frac{\partial}{\partial z}} and {\displaystyle Y=\frac{\partial}{\partial y}+x\frac{\partial}{\partial z}} and define the distribution {D_p=\text{span} \{X_p, Y_p\}}.

Choose a regular value of {f}, say {C} (one exists by say Sard’s Theorem). Then {f=C} is a 2-dimensional submanifold {M\subset \mathbb{R}^3}, and since {f} is a defining function {T_pM=ker(Df_p)}. But the very fact that {f} satisfies, by assumption, {X(f)=0} and {Y(f)=0}, we have {T_pM=\text{span} \{X_p, Y_p\}}. I.e. {M} is an integral manifold for the distribution {D}. Thus {D} must be involutive.

Just check now. {\displaystyle [X,Y]=2\frac{\partial}{\partial z}}, so in particular at the origin {\displaystyle X_0=\frac{\partial}{\partial x}} and {\displaystyle Y_0=\frac{\partial}{\partial y}} it is not in the span, and hence not involutive. Thus no such {f} exists. This didn’t even use Frobenius.

Now let’s spice up the language and difficulty. Is it possible to find a function {z=f(x,y)}, {C^\infty} in a neighborhood of {(0,0)}, such that {f(0,0)=0} and {\displaystyle df=(ye^{-(x+y)}-f)dx+(xe^{-(x+y)}-f)dy}? Alright, the {d} phrasing is just asking there is a local solution to the system {\displaystyle \frac{\partial f}{\partial x}=ye^{-(x+y)}-f} and {\displaystyle \frac{\partial f}{\partial y}=x^{-(x+y)}-f}. Uh oh. The above method fails us now since it isn’t homogeneous.

Alright, so let’s extrapolate a little. We have a system of the form {\displaystyle \frac{\partial f}{\partial x}=\alpha(x,y,f)} and {\displaystyle \frac{\partial f}{\partial y}=\beta(x,y,f)}. The claim is that necessary and sufficient conditions to have a local solution to this system is {\displaystyle \frac{\partial \alpha}{\partial y}+\beta\frac{\partial \alpha}{\partial z}=\frac{\partial \beta}{\partial x}+\alpha \frac{\partial \beta}{\partial z}}.

I won’t go through the details of the proof, but the main idea is not bad. Define the distribution spanned by {\displaystyle X=\frac{\partial}{\partial x}+\alpha\frac{\partial}{\partial z}} and {\displaystyle Y=\frac{\partial}{\partial y}+\beta\frac{\partial}{\partial z}}.

Then use that assumption to see that {[X,Y]=0} and hence the distribution is involutive and hence there is an integral manifold for the distribution by the Frobenius Theorem. If {g} is a local defining function to that integral manifold, then we can hit that with the Implicit Function Theorem and get that {z=f(x,y)} (the implicit function) is a local solution.

If we go back to that original problem, we can easily check that the sufficient condition is met and hence that local solution exists.

I had one other neat little problem, but it doesn’t really fit in here other than the fact that solutions to PDEs are involved.


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