A BOUNDARY LAYER RESULT FOR AN ra-DIMENSIONAL LINEAR ELLIPTIC EQUATION

Introduction. Since the publication of Levinson's results concerning the first boundary value problem for eAu+A(x,y)ux+B(x,y)uy + C(x, y)u = D(x, y) [6], there have been a number of other papers dealing with related two-dimensional problems [l; 3; 7]. More recently a paper by O. A. Ladyjzenskaya has appeared in which an ra-dimensional extension of the original problem is discussed [4].2 It is felt that the following treatment of the ra-dimensional problem making use of a previously devised modification of Levinson's approach [l; 3], may be of interest. It is readily seen that this treatment can be modified so as to include the case of certain types of singularities in the field of characteristics of the e = 0 equation [7]. Roughly speaking, our object is to prove that the solution u(x, e) of a certain e-parametered elliptic equation (1.1) may be characterized as the superposition of two functions. One of the two is a function U(x) independent of e, while the other function z(x, e) is of significant magnitude only near a boundary of the domain. That is, z(x, e) is a "boundary layer" term. The plan of the discussion is as follows. The hypotheses and the geometrical aspects of the problem are discussed in §1. In §2 the formal statement of the theorem is given and boundary layer terms are constructed. Finally, in §3 the desired conclusion is proved by appeal to a maximum principle.