ABSTRACT
The Dirichlet problem, or the first boundary value problem, can be stated as follows: Let f be a given continuous function defined on the boundary of an n-dimensional domain G; find a function u continuous inside G and on its boundary, satisfying the Laplace equation. In the one-dimensional case, the Dirichlet problem is trivial: on any finite interval we can construct a linear function with the given values at the end-points of the interval. Using a construction proposed by S. N. Bernstein, in combination with Perron’s notion of a ‘barrier’, one can prove that the harmonic function in G constructed by the finite difference method takes the values of a given continuous function at all regular points of the boundary. Lichtenstein, Brelot, Giraud published a series of papers on solving the first boundary value problem for elliptic equations of type (2) with coefficients having discontinuities at single points or on some curves or surfaces.
