Can the "physical argument" for the existence of a solution to Dirichlet's problem be made into an actual proof?

Question

Caveat: I don't really know anything about PDEs, so this question might not make sense.

In complex analysis class we've been learning about the solution to Dirichlet's problem for the Laplace equation on bounded domains with nice (smooth) boundary. My sketchy understanding of the history of this problem (gleaned from Wikipedia) is that in the 19th century everybody "knew" that the problem had to have a unique solution, because of physics. Specifically, if I give you a distribution of charge along the boundary, it has to determine an electric potential in the domain, which turns out to be harmonic. But Dirichlet's proof was wrong, and it wasn't until around 1900 that Hilbert found a correct argument for the existence and uniqueness of the solution, given reasonable conditions (the boundary function must be continuous, and the boundary really has to be sufficiently smooth).

Is the physical heuristic really totally meaningless from a mathematical point of view? Or is there some way to translate it into an actual proof?

Answer 1

Well, I don't understand the electrostatics, but here is another physical heuristic:

Impose a temperature distribution at the exterior, and measure (after some time has passed) the temperature in the interior. This gives a harmonic function extending the exterior temperature. [What's the electrostatic analogue? Formerly I had written "charge density", but now I am not sure if that's right.]

I think this strongly suggests a mathematically rigorous argument: We are naturally led to model the time-dependence of temperature in the interior. This satisfies a diffusion (or heat) equation, but in words:

"After a time \delta, the new temperature is obtained by averaging the old temperature along a circle of radius \sqrt{\delta}."

This process converges under reasonable conditions, as time goes to infinity, to the solution of the Dirichlet problem. Anyway, we are led to the Brownian-motion proof of the existence, which I personally find rather satisfying. Another personal comment: I think one should always take "physical heuristics" rather seriously.

[In response to Q.Y.'s comments below, which were responses to previous confused remarks that I made: neither the electric field nor the Columb potential is a multiple of the charge density on the boundary: the former is a vector, and in either case imagine the charge on the boundary to be concentrated in a sub-region; neither the electric field nor the potential will be constant outside that sub-region.]

Answer 2

The physical concept that "prescribing the potential at the boundary should determine the potential inside" does not even suggest a proof. (Btw, I think you meant "potential at the boundary" rather than "distribution of charge along the boundary.") At most it suggests a theorem. However, the physical principle that nature tries to minimize a quantity (in this case, the energy) does suggest an idea for a proof. This is precisely what Dirichlet (and others) tried to do in the 19th century, as well as what was eventually carried out by Hilbert.

To add to what Aguirre wrote, the basic problem is that just because you have an inf does not mean that you have a min: the problem is a lack of compactness. The technical fix is to weaken your topology (while also completing it) to get a larger space of functions in which you can successfully extract a min. The cost is that you no longer know if the min is an honest-to-goodness solution in the usual sense, so you have to work a little to prove that it is. (In other words, you've made the existence question easier at the expense of introducing a regularity question.)

A reference for this method of solving the Dirichlet problem (sometimes called the "direct method" in the calculus of variations) is Rauch's PDE book, which covers this material with a minimal amount of fuss.

Answer 3

The electrostatic intuition does lead to a correct mathematical formulation of the Dirichlet problem.

Let's consider an electric charge distribution of two thin layers (one layer is positive and the other is negative) located along a closed surface $S\subset\mathbb R^3$. Assume that $d>0$ is the distance between charges along the normal $n_p$ to the surface at point $p$. Let $\rho\in C(S,\mathbb R)$ denote the distribution's density.

A pair of two opposing charges $+Q=\rho/d$ and $-Q=-\rho/d$ creates an electric field. The limit of the field when $d\to 0$ is known as the dipole. For any $x\in \mathbb R^3$, the dipole potential at the point $p\in S$ has the form $$\frac{\rho}{d}\Phi(x-(p+n_pd/2))-\frac{\rho}{d}\Phi(x-(p-n_pd/2))=\rho\frac{\partial \Phi(x-p)}{\partial n_p} +o(1)\quad{\rm as\ \ } d\to0,\qquad(1)$$ where $\Phi(x)=-(4\pi|x|)^{-1},\quad x\in \mathbb R^3,$ is the fundamental solution of Laplace's equation. (1) gives the dipole potential of a single dipole at the point $p\in S$ and the integral $$u(x)=\int_{S}\rho(p) \frac{\partial \Phi(x-p)}{\partial n_p} dp,\quad x\in\mathbb R^3,$$ is the potential of the whole distribution.

Now, a simple computation shows that $u(x)$ is a harmonic function when $x$ is not on the surface. It has a jump when $x$ passes through $S$: $$u_{-}(x_0)=u(x_0)-2\pi \rho(x_0),\quad u_{+}(x_0)=u(x_0)+2\pi \rho(x_0),\quad x_0\in S,\qquad\qquad\qquad\qquad (2)$$ where $u_{-}(x_0)$ ($u_{+}(x_0)$) is the limit from the interior (exterior) of the surface.

Relations (2) are integral equations w.r.t. the unknown density (assuming that the potential on the surface is known). The equations can be solved using the Fredholm approach. The function $u(x)$ then gives a solution to the Dirichlet problem.

Edit. See a nice little textbook by Arnold where he shows how to make the physical intuition rigorous in this problem.

Answer 4

Physical intuition tells us that the potential determined by a distribution of charge on the boundary will minimize the energy. If D is a bounded domain in R^d (d=3 in real applications), then the energy of a potential u on D is given by Dirichlet's integral:

DI(u) = \int_D |grad(u)|^2 dx

Thus, if we minimize DI(u) among all functions u whose restriction to the boundary of D is equal to a certain function f, we will have solved Dirichlet's problem: find u: D -> R such that

Laplacian(u) = 0 in D, u restricted to the boundary of D = f.

The Dirichlet integral is bounded below (by zero), and so has an infimum when computed on functions whose restriction to the boundary of D is equal to f. What is not obvious at all is that it attains its minimum. The proof of this fact had to wait until the development of what we now call Hilbert space methods. In particular, one of the problems was to determine the right class of functions to work with: hence the emphasis on all functions above.

Once the existence of a solution is proved, uniqueness follows from the maximum principle.

Answer 5

I am not an expert on PDEs, but I know that often the existence and uniqueness of solutions to partial differential equations are obtained by Banach's fixed point theorem or similar results. Essentially, the idea is that if solutions to a differential equation are "more stable" then the boundary conditions, you should be able to construct a unique solution by a limit process.

If you look into this problem, it very much looks like one of the problems solvable this way. What reinforces my opinion is that you're actually asked to find something in a zero space for a Laplace operator and this is a very good operator, it's elliptic I think, so, e.g. the exp (-t\Delta) expression behaves well and has projection to the zero space as limit.

Hope that gives some direction and keywords to search :)

Answer 6

Do you mean that you want to rigorously prove a mathematical statement on the basis of physics, or that you want a proof based on the mathematical theory of a particular physical phenomenon? I assume you have to mean the latter.

The Dirichlet problem can be solved by the mathematical theory of Brownian motion, which describes a physical phenomenon discovered in the nineteenth century.

But I don't know any proof based on electromagnetic theory (except tautologically, by incorporating the theory of harmonic functions into electromagnetism).