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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.

user118503
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1 Answers1

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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.

fedja
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  • I don't understand where you make use of the harmonicity of $u_{i}$ in your example, as if we neglect it, any smooth extension of $\varphi _{1}$ transfers to a topologically identical extension of $\varphi _{2}$, due to the similarity of the boundary conditions. – user118503 Dec 13 '17 at 06:45
  • @user118503 Harmonic extension feels the "mass", not just outstanding individual values (max/min) and that is the card I'm playing here. An arbitrary smooth extension does not. – fedja Dec 13 '17 at 22:54