262 Difference Schemes for Elliptic EquationsCorollary 2 The homogeneous equation (1) subjecl to the boundary con-
dition(9) £y(P) = 0 on w, y(P) = 0 on /,
has only the trivial solution y(P) - 0.
It is straightforward to verify that y(P) = 0 is a solution of problem
(9). Further, assume to the contrary that there exists a solution y(P) "f:. 0
of problem (9). If y(P) -::j:. 0 at least at one point, then by Corollary 1 the
inequalities y(P) < 0 and y(P) > 0 must hold simultaneously. But it is
possible only if y(P) = 0. Thus, we have proved the following assertion.Corollary 3 Problem (1)-(4) possesses a unique solution.- Comparison theorem. The 1najorant.
Theorem 2 Let y(P) be a solution ofproble1n (1)-(4) and let Y(P) be a
solution of the problem(10) £ Y(P) = F(P), p E w, Y(P) = p(P), PE/.
Then the conditions( 11) I F(P) I< F(P), p E w, I p(P) I< p(P), p E,,
provide the validity of the inequality
(12) I y( P) I < Y ( P) for P E w + /.
Proof By Corolla_ry 1 the inequality Y(P) > 0 is valid on w +I· The
functions u(P) = Y(P)+y(P) and v(P) = Y(P)-y(P) solve the equations
£ u = Fu = F + F > 0 and £ v = Fv = F - F > 0 subject to the boundary
conditions ul,, = (Y + y)I,, = P + p > 0 and vi,, = (Y - y)I,, = p - p > 0.
Since the conditions of Corollary 1 are satisfied, we have u > 0 or
y > -Y, v > 0 or y < Y. It follows from the foregoing that -Y < y < Y
or I y( P) I < Y ( P) on w + /.
The function Y ( P) is called the inajorant for a solution of problem
(1 )-(3). Its determination entails immediately the validity of the desired
estimate for 11 y lie"