Timeline for Geometrical structure of critical points of harmonic functions
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
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| Oct 25, 2010 at 2:41 | vote | accept | Janus Wesenberg | ||
| Oct 23, 2010 at 13:18 | comment | added | BS. | I found an example, see the answer below. | |
| Oct 23, 2010 at 13:17 | answer | added | BS. | timeline score: 5 | |
| Oct 23, 2010 at 12:35 | comment | added | BS. | @Janus : I found this reference on real analytic functions S. Krantz, H. Parks, A primer of real analytic functions. Birkhäuser Verlag, 1992. It was reedited in 2002. As to the "last statement", I mean the first option. I don't know if curves like $y^2=x^3$ can appear in the critical set of a 3-dimensional harmonic function (they can't appear in 2d). | |
| Oct 22, 2010 at 16:17 | comment | added | Janus Wesenberg | @BS Thank you very much -- this is getting very close to what I was looking for. Could you recommend a good reference for somebody with only rusty undergrad analysis and differential geometry as background? Do I understand your last statement correctly that your argument does not guarantee that the curves extend smoothly across the singular points, or is it clear that there are situations in which this the curves would not extend smoothly across the singular points? | |
| Oct 22, 2010 at 8:20 | comment | added | BS. | Since the laplacian is elliptic with real-analytic coefficients, a harmonic function $f$ is real-analytic in its domain of definition. Hence the set $C$ of critical points of $f$ is a real-analytic subset of $\mathbb{R}^3$, and as such it admits a locally finite partition into real-analytic locally closed smooth submanifolds. Thus if $\dim C \leq 1$, it is locally a finite union of analytic open arcs and singular points (but the curves might not extend smoothly across those points). | |
| Oct 22, 2010 at 7:21 | comment | added | Janus Wesenberg | @Will Jagy: Yes, it is certainly possible to construct intersections of analytic curves in this way, although some constraints are imposed by the harmonicity requirements (see arxiv.org/abs/0802.3162 for the case of two curves). What my post pertains to is in some sense whether this approach is complete: i.e. whether all possible intersections of guide curves are intersections of analytic guide curves. You might have to spell out a bit the implications of $\vec{C}\cdot\nabla$ commuting with $\nabla^2$ :) | |
| Oct 22, 2010 at 7:01 | history | edited | Janus Wesenberg | CC BY-SA 2.5 |
deleted 2 characters in body
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| Oct 22, 2010 at 7:00 | comment | added | Janus Wesenberg | @Willie Wong: Thank you very much for pointing this one out. I have corrected my definition of guide curves so it now hopefully describes what I am looking for. | |
| Oct 22, 2010 at 6:54 | history | edited | Janus Wesenberg | CC BY-SA 2.5 |
Corrected definition of critical curves
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| Oct 21, 2010 at 18:40 | comment | added | Will Jagy | In analogy with the imaginary part of $(x + i y)^n$ being zero along $n$ lines through the origin in the plane, my guess is that one can arrange guide curves as lines through the origin in $R^3$ and the vertices of a regular polyhedron, not just the octahedron as you point out. Furthermore I thing harmonic polynomials suffice to do this. What seems less clear is larger numbers of points on the unit sphere. Note that for fixed constants $A,B,C$ the Laplacian commutes with $$ A \frac{\partial}{ \partial x} + B \frac{\partial}{ \partial y} + C \frac{\partial}{ \partial z} $$ | |
| Oct 21, 2010 at 9:39 | comment | added | Willie Wong | You need to be a lot more careful with your quantifiers. Let $\Phi$ be a harmonic function without a critical point, say $\Phi(x,y,z) = x$. then trivially any curve $\gamma$ has the property that for every point $p\in \gamma$, for any neighborhood $V$ of $p$, all critical points of $\Phi$ in $V$ (which comprise the empty set) is in $\gamma$. Now, I am not sure what you mean by "analytical parametrization", but given that ALL curves $\gamma$ are allowed in the above example, if there exists curves that does not have analytical parametrization, then you can draw the obvious conclusion. | |
| Oct 21, 2010 at 7:28 | comment | added | Janus Wesenberg | Background: The question is relevant to networks of rf ion traps, where the trapping potential is the ponderomotive potential associated with an oscillating electric field. The local amplitude of electric potential oscillations is described by a harmonic function, and for practical reasons it is preferable to trap ions on critical points of this function so that transport would have to take place along guide curves as introduced above. The aim of my question is to establish what intersection topologies are possible for guide curves. | |
| Oct 21, 2010 at 7:28 | history | asked | Janus Wesenberg | CC BY-SA 2.5 |