I posed a problem about the solution of the heat equation where temperature along the boundary of a 2D plate was discontinuous. I solved the problem approximately with a radial basis function approach in the computer algebra system Maxima, and got some nice graphs. Around about the time I got some nice graphs, Maarten asked me: "Are you sure the solution exists?". For about a half hour or so we stood around in front of a whiteboard and discussed whether I was fooling myself with my numerical technique, or was it indeed converging to something real?

I believe I have an informal proof that the solution exists even though the gradient and laplacian do not.

The proof goes something like this:

The steady state heat equation can be written in non-dimensional form as $\frac{\partial^2 T}{\partial x^2} +\frac{\partial^2 T}{\partial y^2} = 0$ for $(x,y)$ inside the domain, and in my case $T(x,y) = f(x,y)$ along the boundary with $f$ being a step function, so that $T=0$ along the square part, and $T=1$ along the curved part.

The Feynman-Kac formulation of the heat equation relies on the fact that there is a connection between brownian motion of a particle, and diffusion of heat. Basically, you can think of temperature as a concentration of energy. Photons come from the boundary and diffuse into the interior, and then back out of the boundary. If the temperature at x is at steady state, then for any $dt$ the expected change in concentration of energy $dT=0$. Now if we run time backwards, we can say that the photons coming into the point x are now going out of the point x, and they randomly walk around inside the plate until they exit the boundary. Where they exit tells us how much energy they brought to the mix (based on the temperature on the boundary).

So in my situation, the Feynman-Kac formulation shows us that the temperature inside the plate is the expected value of the exit temperature, which is a random variable that takes on the values of 0 or 1. Since we're taking the expectation of a random variable that can't be outside the interval [0,1] the expected value is between those extremes. This means that at steady state, for every point inside the boundary, the temperature is less than 1 and greater than 0. Since that's true, the function that defines the temperature has finite 2 norm, and is a perfectly nice function, except right at the two step locations along the boundary. Since the probability that a photon exits exactly at the step location is 0, we have no problem with those points in the Feynman-Kac formulation.

So, yes, the heat equation has a solution even though for that solution $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}$ does not exist at some points.