ABSTRACT
The general procedure which Hadamard has called “the method of descent” (“la méthode du descente”) has been employed by Boudjelkha and Diaz (see [2] and [3] of the Bibliography at the end of the paper), using as a starting point the known solution, by means of Poisson’s integral, for the Dirichlet problem for a half space for the partial differential equation of the second order (Laplace’s equation) in n+1 real dimensions: Δ n + 1 w ≡ ( ∂ 2 ∂ x 1 2 + ∂ 2 ∂ x 2 2 + … + ∂ 2 ∂ x n 2 + ∂ 2 ∂ x n + 1 2 ) w = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072577/94f683e7-9f6e-41e2-9c5e-98432cdb82df/content/eq422.tif"/>
in order to obtain, after the “descent,” the solution of the Dirichlet problem for a half space for “the λ equation,” also of the second order, in n real dimensions, Δ n u − λ 2 u = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072577/94f683e7-9f6e-41e2-9c5e-98432cdb82df/content/eq423.tif"/>
where λ is a real constant.
The consideration of these results, which have just been recalled, suggests the following natural question: What sort of results can be obtained if, instead of starting initially, in applying the method of descent, with the second order equation of Laplace Δn+1 w=0, 60in order to descend to the “λ equation,” also of the second order, Δnu – λ2u = 0, one takes as a starting point the fourth order equation, the so-called biharmonic equation, that is to say, Δn+1,(Δn+1w) = 0? The purpose of the present paper is to give an accurate partial reply to this intuitively formulated question. To fix the ideas, and for clarity and simplicity, only the particular case of n = 2 is carried out in detail.
The principal difficulties encountered in the course of the completion of this research were:
The solution, by means of explicit formulas, of the boundary value problem for the biharmonic equation in a half space, when the unknown function and the normal derivative of this function are prescribed on the boundary. In order to attain the purpose indicated above, it was found necessary to solve this classical problem more precisely than it is solved in the usual treatises, and the exact result, which is here obtained, does not seem to appear elsewhere in the mathematical literature.
After having solved the boundary value problem for the biharmonic equation, which has been mentioned in (i), it was necessary to apply the method of descent to the biharmonic equation, to solve, in this way, the corresponding boundary value problem for a certain “λ equation.” Another difficulty arose, at this particular point, in the research, because, a priori, it did not seem possible to predict, with certainty, exactly what would be the “fourth order λ equation” (as a sort of proof of this assertion, we propose to the reader to try to guess the precise form of the “fourth order λ equation,” before reading Theorem IV.3 of Chapter IV).
