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This was a question that arose for me when I was thinking about how one proves strong unique continuation for elliptic equations. I never could come up with a satisfactory answer.

Background: When one wants to prove strong unique continuation results for elliptic operators (i.e. the only solutions that vanish to infinite order are the zero function) it is usually the case that one shows more quantitative doubling estimates. However, the constants in these estimates depend on the solution under consideration (in a reasonably explicit way). I was trying to understand this better for myself and so was considering the following question.

The (simplified) setup is as follows: Suppose that $\mathbb{D}$ is the unit disk in $\mathbb{R}^2$. Let $u_i$ be a sequence of harmonic functions on $\mathbb{D}$ and suppose that $u_i\to 0$ uniformly on compact subsets of $\mathbb{D}$.

Examples:

1) $u_i \equiv \frac{1}{i}$. (Okay this one is dumb)

2) $u_i = \frac{1}{i} H_n(x,y)+\frac{1}{i^2} H_m(x,y)$ where $H_n(x,y), H_m(x,y)$ are fixed harmonic polynomial of degree $n$ and $m$.

3) $u_i= \frac{1}{i} H_i (x,y)$ where $H_i(x,y)$ is a sequence of (EDIT: homogenous) harmonic polynomials of degree $i$ and normalized so $\sup_{\mathbb{D}} |H_i|=1$.

4) $u_i=\frac{1}{(i!)!}+\frac{1}{i} H_i (x,y)$ where $H_i$ are as in 3) also (i!)! is just supposed to be a really big number.

Now 1) and 2) are qualitatively different from 3) and 4). In particular, for 1) and 2) one could hope to ``model" the way the $u_i \to 0$ (by $\frac{1}{i}$ and $\frac{1}{i} H_n$ respectively) whereas for 3) this seems to hopeless (unless you took such a sequence as its own model). However, for 4) it becomes very unclear what the right model would be...

My questions are:

A) Is there a good way to distinguish between sequences like 1) and 2) versus those like 3)?

B) If so is there a good method to extract the ``model"?

C) What sorts of extra conditions could I add to be able to say anything? (e.g. $u_i>0$ helps a lot).

D) Is this hopeless?

Things I've thought:

a) If the $u_i$ are all positive, then one can normalize by dividing by $u_i(0,0)$ and using a Harnack inequality and Arzela-Ascoli to get a good model. This is a really strong condition though...

b) It seems that the frequency functions of Almgren might be a natural way to recognize 3) but I worry something like 4 introduces difficulties of this approach.

Any thoughts would be appreciated!

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