Assumption of Boundary Values and the Green's Function in the Dirichlet Problem for the General Linear Elliptic Equation.

During the past year, the author has presented in two notes in these PROCEEDINGS a general method of solving boundary value problems for linear elliptic differential equations of arbitrary order which combines essential features of the integral equation method and Dirichlet's Principle or the method of orthogonal projection.1 In particular we have shown that the Fredholm alternative holds for the solutions of the Dirichlet problem in an arbitrary bounded domain where the boundary values are to be taken on in the generalized sense familiar from the work of Courant, Friedrichs, and others on the variational approach to elliptic differential equations.2 It is the first object of this note to outline the proof in the case of two independent variables that for a smoothly bounded domain, smooth boundary data, and an equation with strongly differentiable coefficients, the solution of the Dirichlet problem actually assumes the full set of boundary data in the classical sense. Similar results in n independent variables are derived by the same method. However, for two independent variables, no a priori smoothness conditions on the solution need be imposed which are not derived in the general existence theorems.3 Our second object is to establish the existence and properties of the Green's function for an arbitrary bounded domain in En on which the Dirichlet problem has no null solutions. The proof of Theorem 1 relies extensively upon the results and methods developed by Fritz John in his construction of a fundamental solution in the small.4 Our methods are extensible without any alteration in principle to the more general case of. strongly elliptic systems of differential equations.5 Throughout the following discussion, we shall use the notation of our previous notes. 1. Let D be a bounded domain6 in En, C,2(D) the linear space of 2m179 VOL. 39, 1953