Green's function in some contributions of 19th century mathematicians

Abstract

Many questions in mathematical physics lead to a solution in terms of a harmonic function in a closed region with given continuous boundary values. This problem is known as Dirichlet's problem, whose solution is based on an existence principle-the so-called Dirichlet's principle. However, in the second half of the 19th century many mathematicians doubted the validity of Dirichlet's principle. They used direct methods in order to overcome the difficulties arising from this principle and also to find an explicit solution of the Dirichlet problem at issue. Many years before, one of these methods had been developed by Green in 1828, which consists in finding a function-called a Green's function-satisfying certain conditions and appearing in the analytical expression of the solution of the given Dirichlet problem. Helmholtz, Riemann, Lipschitz, Carl and Franz Neumann, and Betti deduced functions similar to Green's function in order to solve problems in acoustics, electrodynamics, magnetism, theory of heat, and elasticity. © 2001 Academic Press.