Herbert Koch (Bonn)¶
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Theorem (K, Nadirashvili): $(M,g), g \in C^1$, harmonic map $u: U \rightarrow M$ and you know that $u \in C^1, Du(x_0) \neq 0 \implies$ local charts are analytic.
Elliptic PDE
Expectation: solutions are as regular as the data.
$F,f$ are analytic.
What does it mean to say the equation is elliptic?
$$a_{jk} = \frac{\partial F^j}{\partial p_k} (x,u, p), ~ a_{jk} + a_{kj} ~{\mbox{positive definite}}.$$(elliptic)
$u \in C^1$ is a weak solution.
Theorem: $u$ is analytic.
This work is closely related to older work by Morrey, by Levy, and others. The reason I want to present it here is I wish to generalize the hypotheses for some other purposes.
Proof: This is a local statement. So, we can translate and subtract off an affine function.
step 1. $u(0) = Du(0) = 0$ rotate, smallness, $a_{jk} (0,0) = \delta_{j,k}$. The last assumption is not strictly correct but we can proceed to resolve things from progress here.
How to think about analyticity?
step 2. $\sum_{j=1}^d \partial_{z_j} F^j (z, u, D_z u) = f (z, u, D_z u)$
complexify
You might expand this out by writing it as $$ \sum_{j=1}^d \partial_{z_j}^2 u = \sum_{j=1}^d \partial_j G^j (z,u,D_Zu ) = f(z,u,D_z u) $$
This equation does not have good structure. But it has another nice feature. The Cauchy-Riemann equations can be found here. $$ \frac{\partial u}{\partial y_j} = i \frac{\partial u}{\partial x_j}$$
Q (Tarantello): Why do you assume all this analyticity? Aren't you trying to prove that?
A: We are first imagining the result is true and then seeing what we can find in terms of the structure of the equation.
"I want to somehow change this into something that is more familiar"
He draws a picture and starts defining some structures associated with the picture. This is a scenario where typing is inadequate relative to writing with a pen!