TABLE OF CONTENTS Let us confirm that equality (5.59) is valid even if u ? C 2( B) is not harmonic, where B = B(0, 1) and ? = B\ B( x, ?) with ? B and 0 < ? ?1.
The left-hand side is equal to
and hence converges to
as ?? 0. On the other hand, the right-hand side accepts the same treatment and hence converges to
Thus, we obtain
The analogous equality to (5.61) is similar, and is given as
Thus, we obtain
with the Green's function
if u ? C 2( B) satisfies
In particular, if u ? C 2( B) solves the Poisson equation
then it is given as
However, deriving (5.86) from (5.85) is not so simple.
First, because B satisfies the outer circumscribing ball condition at any boundary point, any f ? C( ? B) admits a unique u ? C 2( B) ? C( B) satisfying (5.86) with g = 0. By Theorem 5.18, this u( x) is given by
Therefore, if f ? C( ? B) the first term of the right-hand side of (5.85) is in C 2( B) ? C( B) and satisfies
On the other hand, g ? C( B) cannot imply the first term of the right-hand side is in C
