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I'm hoping someone can explain (at the lowest possible level) how PDEs (or evolution equations, giving rise to PDEs locally) are solved on a manifold. For example, the Laplace-Beltrami equation on a closed manifold: $$ \triangle_g f =0.$$ In any coordinate patch there is a local PDE on $\mathbb R^n$ that must be satisfied. I've been told you just look at the PDE in local coordinates and solve that, but how is it usually shown that this defines a global solution? i.e. if we just look at an arbitrary patch and find a solution how does one know that there is actually a single function on the manifold that satisfies the equation or is equal to the local solutions we have shown exist.

I feel I'm missing something fundamental b/c I can't find this addressed anywhere I have looked.

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  • $\begingroup$ The analysis you do to show the existence of a solution is usually (in my experience) global. When you want to show regularity properties of the solution you work locally. $\endgroup$ – user98602 Sep 11 '15 at 21:10
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The answer depends on which type of PDEs you are considering.

Elliptic

The elliptic theory is pretty well exposed in the three-volume work of Michael Taylor on PDEs. A main point to the theory of elliptic PDEs is that you can often split off existence problems, uniqueness problems, and regularity problems.

The regularity problems are in general independent of your "how do you know there is a global solution" question, since regularity is entirely a local phenomenon, and the question asked is "given a solution in some function space $X$, can we say that it is in some better function space $Y$", so the object you are playing with is already defined.

For the uniqueness problems, one sometimes consider global uniqueness of solutions; for example, when dealing with linear elliptic problems on compact domains one can usually apply the maximum principle to gain uniqueness of solutions. In this situation we are dealing, again, with some solutions which we assume to exist and we want to show that they are the same. And again this does not involve the "global" problem you mentioned.

It is usually in the existence step that one usually runs into this problem you are facing. And the basic solution to it is that one deals with existence problems often using sufficiently abstract and global machinery that this is not an issue.

For example, for linear equations one may want to establish existence of weak solutions using functional analytic techniques. The abstract theory of Hilbert spaces are sufficiently agnostic to the underlying manifold you can usually prove the existence of weak solutions on a manifold in pretty much the same way you would on $\mathbb{R}^n$ or domains thereof (there are of course exceptions, where you have for example topological constraints). Alternatively you can also obtain existence using variational techniques (minimize a functional). The key here is that one never actually goes in to "solve the equation locally" when it comes to showing the existence of solutions.

Hyperbolic

For hyperbolic equations on manifolds the answer is rather different. One of the key properties of hyperbolic PDEs is the "finite speed of propagation" property. This allows one to effectively localize even the "existence and uniqueness" parts of proofs (for wellposedness questions concerning the initial value problem) to work in coordinate charts.

The key is that you prove both a local uniqueness and a local existence theorem simultaneously. The rough argument goes something like this:

  1. You want to solve a hyperbolic PDE on the product manifold $I\times M$ where $I$ is an interval representing the time coordinate and $M$ is some manifold representing the space coordinate.
  2. You take some charts $\{U_\alpha\}$ covering $M$.
  3. You should that for any chart $U$ on $M$ you can solve (roughly speaking) the equation on $I\times U$, and that the solution is unique. (This is not strictly true, but close enough to give the picture.)
  4. Now consider $U_\alpha$ and $U_\beta$ with non-empty intersection. You've constructed a solution $f_\alpha$ on $I\times U_\alpha$ and a solution $f_\beta$ on $I\times U_\beta$. These two can be pieced together to get a solution $\tilde{f}$ on $I\times (U_\alpha\cup U_\beta)$ if $f_\alpha|_{I\times U_\alpha\cap U_\beta} = f_\beta |_{I\times U_\alpha\cap U_\beta}$. But this last equality is guaranteed since the restriction of either $f_\alpha$ or $f_\beta$ to $I\times U_\alpha\cap U_\beta$ is a solution to the equation, and by the local uniqueness statement in step 3 any solution is unique.
  5. Rinse and repeat and everything pieces together.

Hans Ringstrom's book on the Cauchy problem in general relativity is a good place to checkout arguments of this type.

Parabolic

For linear parabolic equations on manifolds with no boundary, one often again appeal to abstract nonsense to establish existence of evolution. The keyword here is "semigroups". There is a pretty good discussion on MathOverflow about this.

When you have nonlinear equations, especially on manifolds with boundaries, things can get a bit tricky, if I remember correctly. And I am not sufficiently well-versed in the theory to tell you too much details.

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  • $\begingroup$ Thank you very much! this is just what I was hoping for. $\endgroup$ – Ashley Sep 11 '15 at 22:25

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