Let $\varphi\_{1},\varphi\_{2}:\mathbb{S}^{1}\rightarrow\mathbb{R}$ be two
smooth general position (Morse) functions having the same set of critical
points $\left\{ p\_{1},...,p\_{n}\right\} \subset\mathbb{S}^{1}$ ($n$ is even)
and both $\varphi\_{1}$ and $\varphi\_{2}$ have a local maximum at $p\_{1}$.
Suppose that $\varphi\_{1}$ and $\varphi\_{2}$ are *similar* in the following sense:


$\left( \varphi\_{1}(p\_{i})-\varphi\_{1}(p\_{j})\right) \left( \varphi
\_{2}(p\_{i})-\varphi\_{2}(p\_{j})\right) >0$ for any $i\neq j$,


i.e. the critical level sets of $\varphi\_{1}$ and $\varphi\_{2}$ are in some
sense similar. 


Consider the corresponding two Dirichlet problems:


$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Delta u=0$


$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u|\_{\mathbb{S}^{1}}=\varphi\_{i}$ , $i=1,2$,


getting in such a way two harmonic solutions $u\_{1},u\_{2}:\mathbb{B}%
^{2}\rightarrow\mathbb{R}$.


Then is it true that the level lines portraits of $u\_{1}$ and $u\_{2}$ are the
same up to topological equivalence, i.e. there is a homeomorphism
$h:\mathbb{B}^{2}\rightarrow\mathbb{B}^{2}$ fixing all $p\_{i}$ and sending the
level lines of $u\_{1}$ onto the level lines of $u\_{2}$? Then, of course, $h$
is sending the critical set of $u\_{1}$ onto the critical set of $u\_{2}$.


In brief: does the similarity of the boundary conditions implies similarity
between the solutions of the corresponding Dirichlet problems?


Note that we don't assume $\varphi\_{1}$ and $\varphi\_{2}$ to be close in any sense.



Topological similarity of solutions to Dirichlet problem
Certainly not. Imagine six critical points: $3$ minima and $3$ maxima so that all minima are about $-1$ and all maxima are about $+1$ (but may be all different if you want).
Now assume that the first minimum is "thick", so the corresponding value of about $-1$ spreads almost until the adjacent maxima. If out of the other two one is thick as well and another one is thin (i.e., the function goes to almost $+1$ if you move away just a tiny bit, then, when your level goes up from $-1$ to $1$, the thick sublevel domains of $u$ will meet before the thin sublevel domain will have any chance to spread anywhere. But you are free to choose whom to feed up to thickness and whom to starve to thinness.


