On the Dirichlet problem for linear differential equations

Abstract

admits a unique solution to the Dirichlet problem for sufficiently smooth regions and boundary data. This question also appears to be of some interest in the general theory of partial differential equations. The system (1) is elliptic if and only if (X + 2ju)/*5^0, strongly elliptic if and only if (X + 2p)p>0. When it is strongly elliptic, existence and uniqueness of solutions to the Dirichlet problem for a rather general class of regions and boundary data can be inferred from results of Browder [l] and Morrey [2]. It is noted in [3] that uniqueness fails when ( + 2p)p = 0. For the elliptic systems obtained from (1) by setting n = 2, \= — 3p^0 and replacing u2 by — w2 or «2 by 21/2w2, Bicadze [4] has shown that uniqueness fails. To settle the question of uniqueness, it thus remains to dispose of the case (X + 2p)p<0, which we do in the following