Physical intuition for the Dirichlet problem

Question

I'm studying complex analysis right now, and I am reading about the Dirichlet problem in Greene and Krantz's book Function Theory of One Complex Variable. In Chapter 7, Section 7, the authors mention the Dirichlet problem and its connections to thermal equilibria:

Let $U\subseteq \Bbb C$ be an open set, $U\ne \Bbb C$. Let $f$ be a given continuous function on $\partial U$. Does there exist a continuous function $u$ on $\overline U$ such that $u|_{\partial U} = f$ and $u$ is harmonic on $U$? If $u$ exists, is it unique? These two questions taken together are called the Dirichlet problem for the domain $U$. It has many motivations from physics ... For instance, suppose that a flat, thin film of heat-conducting material is in thermal equilibrium. That is, the temperature at each point of the film is constant with passing time (the system is in equilibrium). Then its temperature at various points is a harmonic function. Physical intuition suggests that if the boundary $\partial U$ of the film has a given temperature distribution $f\colon\partial U\to\Bbb R$, then the temperatures at interior points are uniquely determined [emphasis added]. Historically, physicists have found this intuition strongly compelling, although it is surely not mathematically convincing.

Unfortunately, I don't have the slightest bit of physical intuition for this problem. To me, the boundary of the film and the points of the interior seem to have a very tenuous connection, at least for interior points that aren't too close to the boundary. It seems like I could just change the temperature of one of the interior points very slightly without the boundary knowing anything about it!

$\underline{\mathrm{Q}}$: Can somebody share the physical intuition that the authors say has historically been so compelling? (Feel free to correct the tags on this question, by the way; I wasn't sure how to tag it.)

Answer 1

To me, the boundary of the film and the points of the interior seem to have a very tenuous connection, at least for interior points that aren't too close to the boundary.

I am not sure what you have in mind when you say "tenuous connection", but as far as temperature distribution is concerned, all the points are connected because any real material has finite conductivity and therefore heat transfer can take place between them by virtue of temperature difference between distinct points.

It seems like I could just change the temperature of one of the interior points very slightly without the boundary knowing anything about it!

Not true, unless the conductivity of the material is zero. But any real material has finite conductivity. If the conductivity is small it takes longer for the boundary to "know" that the temperature at the interior point has been changed. If $\alpha$ is the thermal diffusivity, and $L$ is the distance between two distinct points, then a change of temperature of one of those of points, is communicated to the other point on a time scale $L^2/\alpha$. You can see for yourself by dipping a metal and a plastic spoon (the latter has lower $\alpha$) in boiling water and see whose end heats up first.

I must also mention that temperature distribution is harmonic, i.e. it satisfies $\nabla^2T=0$, only when it has reached steady state. Heat transfer is not instantaneous (again, because of finite conductivity). When you change the temperature of an interior point the system is not in steady state anymore. During the evolution towards the steady state, the temperature distribution obeys the (slightly) more complicated equation: $\partial T/\partial t=\alpha \nabla^2T$, in which $t$ is time. Therefore, in a sense, what you say is correct: the boundary will not know for some time that the temperature of an interior point has been changed, but it will know eventually.

Physical intuition suggests that if the boundary $\partial U$ of the film has a given temperature distribution $f\colon\partial U\to\Bbb R$, then the temperatures at interior points are uniquely determined.

Not physical intuition, but real-world experiments suggest that if the boundary $\partial U$ of the film has a given temperature distribution $f\colon\partial U\to\Bbb R$, then the temperatures at interior points are uniquely determined. Physical intuition is no more than accumulated experience from such real-world experiments.

Historically, physicists have found this intuition strongly compelling, although it is surely not mathematically convincing.

I doubt a physicist would say that.