The maximum principle 263Corollary For a solution of the problem(13) Ly= 0 on w, y(P) = μ(P) on /,
the estimate(14) max I y(P) I = II y lie < IIμ lie
PEw+-y Yis valid with IIμ lie.., = maxPE-y I p(P) I·
Indeed, let us specify the majorant Y(P) by the conditions£ Y = 0
on w and Y = II p lie '( on /· The function Y(P) is nonnegative on w + /
and attains its maximum at some node of the grid. This node is none the
inner nodes if Y(P) -::/-const and, hence,II Y lie= PEw+-y max Y(P) =max PE-y Y(P) = IIμ lie ..,.
IfY(P) = const, then Y(P) = llμlle..,· In both cases llYlle = llμlle..,·
Combination of this relation and the inequality II y lie < II Y lie gives esti-
mate (14).- The estimate of a solution to the nonhomogeneous equation. In the
further development a solution of problem (1)-(3) is viewed as a sum
Y=fJ+v,
where y = fJ( P) is a solution to the homogeneous equation
( 15) £ fJ = 0 on w, fJ=μ(P) on/,
and v = v(P) is a solution to the nonhomogeneous equation
(16) £ v(P) = F(P) on w, v(P)=O on/·
We have already obtained estimate (14) for y(P) and so it remains to
evaluate the function v( P).
Theorem 3 If D(P) > 0 everywhere on the grid w, then a solution of
problem (16) admits the estimate
(17) llvlle <II~ lie·