How to solve a variational problem, II

Lots of text today = lots of filler photos. Some ducks from the park next door to Rice.

This is a method used to show that a minimizer exists for a variational problem, though it typically provides no clues as to what the minimizer is.  It is typically referred to as “the direct method in the calculus of variations”.  See yeseterday’s post, at least the first paragraph or two, for background.

First, a point I forgot about yesterday since it is such a convention: “why minimization?”  I would guess this is motivated by physics, and has been explored by Hildebrandt and Tromba in The Parsimonious Universe– roughly, that nature seeks to minimize (rather than maximize) energy spent.  So much of math has been motivated by physics (often useful, as it provides an intuition as to what should be true), it is only natural many of our questions are posed in this manner.  This parsimony can also explain why we typically make sure that the Lagrangian is nonnegative- since energy is nonnegative.

From xkcd. We take the second path here.

Now, the direct method, along with intuition of how we find a least area surface given a prescribed boundary, and what could go wrong in general.  We denote the area of a surface u by F[u]:


Geometry? A bridge in upstate New York?

Show that there is at least one candidate minimizer.  For the least area surface, we pick a point not on the boundary, and draw a straight line from that point to each boundary point to make our first candidate.  As a silly example of what could go wrong, consider trying to minimize the integral of |u|, subject to u(0) = 1 and u(0) = 2.  Well, we just cannot satisfy that boundary condition with any function.

2. Take an infimum.  We now have a nonempty set of candidates \{ u_{\alpha} \}, which is associated with a nonempty set of real numbers \{F[u_{\alpha}]\}.  Going back to the physics explanation, I’ll assume that the integrand is bounded below by zero, and invoke a property of the real numbers to say that there is a greatest lower bound.  It will be our goal to find a surface realizing this infimum.  Notice though that this step is not constructive, so right here we lose our chance at finding a minimizer analytically.  Really the only thing that could go wrong is if our integrand is not bounded below.  It would be somewhat silly to try to find the function whose integral is the “most negative”.

Once we have an infimum, M, we can take a sequence of candidates \{u_j\} which converge to the infimum, F[u_j] \to M.  Notice that the candidate functions do not necessarily converge to anything.  Yet.

"Bikeman" in some of the most beautiful woods in the world.

3. Invoke compactness.  Now, lurking under the surface has been this “space of admissible functions”.  Often this is a Sobolev space, which has the property that the unit ball is “(weak-*) compact”.  See this short paper (pdf link) by Frank Morgan on compactness, which he called “the most important concept in mathematics”.  Compactness implies that every bounded sequence has a convergent subsequence.  Here “bounded” means “with respect to the norm on the space”.  In the case of area minimizers, there is a “mass norm” on the admissible functions (“integer rectifiable currents”) which measures the surface area, and a famous (for some value of the word “famous”)  compactness theorem for these functions (defining a space of admissible functions that was physically relevant and so that such a theorem existed was, perhaps, the most difficult part of Plateau’s problem).  Hence, we now have a subsequence \{u_{j'} \}, and a target (admissible) function u so that u_{j'} \to u and F[u_{j'}] \to M, still.

A sequence of functions converging to the infimum for the integral of |u(x)| subject to u(0) = 0, u(1) = 1. However, there is no smooth function that they are converging to. This sequence appears on the cover of Frank Morgan's real analysis text.

Last thing to show is that F[u] = M.  The two conditions we’ve established certainly nod suggestively at this last equation, but don’t give it themselves.  The figure at right shows what could go wrong.

4. Lower semicontinuity of the functional. We need to show that F[u] = M.  A typical approach is to prove that the functional is lower semicontinuous.  That is to say, F[\liminf u_j] \leq \liminf F[u_j] (fun story: when I asked my advisor to define “upper semicontinuity”, he gave me the finger.  Bonus points if you can see why this is a reasonable definition).  If this is true, then we have M \leq F[u] = F[\liminf u_j] \leq \liminf F[u_j] = M, and are done.  The surface area is lower semicontinuous, which is how we complete the proof of the existence of area minimizing surfaces.

Reusing figures! Non lowersemicontinuity.

The Takagi curves from yesterday provide an example of a functional which is not lower semicontinuous, since F[u_j] \to 0, but F[u] = 1.


Higher resolution

So my normal workflow for creating images here is to sketch ideas, try to create them in MATLAB, and then annotate them in Skitch, if necessary (or sketch things directly into Skitch).  Both these programs are fantastic for typical image manipulation and creating diagrams, though Skitch is (necessarily) a little underpowered.  I have run into this problem once before, where Skitch will take screenshots off your computer (from, for example, a webpage, or a MATLAB figure window), but you cannot layer multiple screenshots into the same image without working a little too hard.  Also, yesterday’s post had some pretty terrible resolution images that I produced in Skitch by trying to just put a textbox in, and then save it.

The 1000th line. Click for full resolution!

The problem with both of these is that I crossed a line into more serious image editing- creating layers, or looking for very high resolution.  I finally worked out that open-source Gimp (and I would assume it’s commercial analogue, Photoshop), a program intended for this sort of image manipulation, is a better choice.  Hence I can offer today a high res version of the 1000th line (1.3MB) and 2000th line (3.1MB) of Pascal’s triangle.  The 1000th is 6pt font, and usually readable with compression.  The 2000th line is 5pt font (otherwise the canvas was too big… it is a 100MB+ file before compression), and only sometimes readable.

The 2000th line. Click for higher resolution!

Compare this to Wikipedia’s image that helped to inspire this experiment, which they display by coloring the pixels in a grayscale from 1 to 10, one pixel per digit.  This has the benefit of higher compression, since we do not need to keep track of the value of each pixel, and easier reading from a macro level.

From Wikipedia.

Finally, just to keep things a little original, here is a similar plot of the first 2000 terms of the Fibonacci sequence.  Again, the font is pretty small, but it gives a good idea of a famous relationship, which is that for large numbers, the nth Fibonacci number is close to \phi^n/\sqrt{5}, where \phi = \frac{1+\sqrt{5}}{2} is the golden ratio.  This is an easy and beautiful derivation using eigenvalues in linear algebra.  But for now, just notice that the number of digits (that is, the log of the number) looks like it increases linearly.  This could lead us to try to prove that \log{F_n} \approx k\cdot n, where F_n is the nth Fibonacci number, and k is just some number.  It turns out we would find that \log{F_n} \approx \log{\phi}\cdot n-\log{\sqrt{5}} which, while sort of ugly, is a linear function in n.

The first 1000 Fibonacci numbers, written vertically. Click for full resolution.