Why does food microwave more evenly if you organize it in the shape of a torus? Any lover of leftovers knows that if you push your food to the edge of a plate and form a donut shape before microwaving it, the food heats up much more evenly and you don't get the "cold middle". I know very little about physic; my peanut sized math major brain is telling me this has something to do with the heat equation on the torus given by
$$\partial_t u(x,t)+(-\Delta)^{\alpha/2}u(x,t)=0, u(x,0)=u_0(x)$$
where $u(x,t)\in \mathbb{T} \times \mathbb{R}$ and $\alpha >0$. This, of course, isn't just a sum of second partial derivatives like the classical heat equation. Anyways, is there a good physics story for why this happens? I'm not sure how to think about this due to a lack of knowledge on physics.
 A: I am by no means a physist however to my understanding the uneven heating is more indicative of the microwaves themselves. They oscillate at a set wavelength and frequency which then is absorbed by the food. These waves bounce around the microwave and dont lead to a uniform heat pattern. There also is the question of food composition and simply the composition of the microwave oven itself. Liquid water is likely going to excite far more quickly and heat up faster than solid ice for instance. So a cake with a liquid interior may heat differently than say a dry piece of cake with an outer frosting. Additionally, a thin evenly spread "donut-shape" as you put it is less dense with more surface area to interact with the microwaves.  A rotating tray may also distribute the microwaves differently than a stationary one. Sorry if my answer is a bit convoluted but I have a peanut sized biology brain.
A: Here's what I think, but I could be wrong.
The wavelength of the radiation in microwave ovens is actually quite large, you can see this yourself: the frequency (which you should be able to obtain from the back of the machine) is around 2.45 GHz which leads to a wavelength of around 12 cm. The microwave oven sets up a standing wave in the cavity, with a wavelength of around 12 cm. Now, this wave will not end up heating the food you place inside it evenly: the food placed at the antinodes (where the wave has, and therefore transfers, the most "energy") will heat up more than the food placed at the nodes (where the electromagnetic field strength is zero, i.e. the red dots below).

(From Wikimedia Commons)
I would imagine that in order for food to cook evenly, you would like to avoid placing part of it at a node, and making a ring seems to be a quick way to do that, given that the nodes will be around 6 cm apart (half a wavelength) and this is more or less the inner diameter of the average "donut" shape you could form on your plate. Avoiding nodes would avoid cold spots, leading to a more even heating profile.
Incidentally, you can see where the antinodes are in your own microwave, chocolate melts more at these points, and there are a bunch of possibilities for nice experiments with this: using chocolate or marshmallows to measure the speed of light. It's a fun exercise to do with your kids (if you have any) or just by yourself, if you need an excuse to buy chocolate. ("It's for science!")

EDIT: So in looking up @BobD's comment below, I found a really nice site by Greg Blonder at Boston University that's basically done extensive analysis on this here. I had earlier thought that the "donut" shape appeared naturally because the same standing wave is rotated along the $z-$ axis. An image from the site showed me that this isn't true, the standing waves look more like this (the wavelengths are still around 12cm though, just diagonal):
 
However, when they are rotated, a central cold spot seems to appear. So it seems that the rotation is an important factor on why the "donut" shape is ideal. I'm including an image from the site below.

(The above images are from Greg Blonder's site)
