Henri Poincaré: Solving Dirichlet’s problem

Dirichlet’s problem reduces to the search for Green’s functions.

Let’s find a function V which inside a surface S satisfies Laplace’s equation and on this surface takes given values.

Suppose that we have found a function U which is finite, which satisfies the equation ΔU = 0, continuous inside S except at the point (a, b, c) inside S where the function U will be infinite, from such a way that the difference

U – 1/√((x – a)² + (y – b)² + (z – c)²)

be finished. To complete the definition of the function U, we will subject it to annihilation at all points of S. We then see that the value of V at the point is equal to the integral

∫(V dV/dn)/4π dω

extended to all elements dω of S.

So when we know how to find Green’s functions, we will know how to solve Dirichlet’s problem.

On the other hand, the search for Green functions is reduced, using the transformation by reciprocal ray vectors, to the problem of electrostatic distribution on the surface of a conductor.

This problem of electrostatic distribution is only a particular case of Dirichlet’s problem, and yet we see that the general case comes down to it. Murphy’s alternating method also makes it possible to reduce the problem of the distribution to the surface of several insulated conductors, in the case of a single conductor. From now on, we will therefore only concern ourselves with the particular case where we are looking for the electrostatic distribution at the surface of a single conductor, since the general case 1 leads back to it.

What should we think of the methods proposed so far? They are both demonstration methods intended to establish the possibility of the problem and computational methods intended to actually solve it. As methods of demonstration, they are quite complicated, but they complement each other in such a way as to apply to all cases and to satisfy judges who are more severe about rigor.

As methods of calculation, they are worthless; because no one will ever have the idea of applying them; even the simplest of them, those of Neumann or Robin, lead to inextricable calculations from the second approximation. All that can be hoped for, without too overwhelming labor, are some rather gross inequalities to which the conductor’s capacity will be subjected.

Source: Henri Poincaré, Sur les Equations aux Dérivées Partielles de la Physique Mathématique, American Journal of Mathematics, Mar., 1890, Vol. 12, No. 3 (Mar., 1890), pp. 211-294. Translation by Nicolae Sfetcu. © 2023 Nicolae Sfetcu