In the present section we consider Neumann problem (1), (2) with .
be the following matrix
where , . Note that if . Denote by , , the determinant of the matrix obtained from A by deleting the elements of the first column and the j th row. Obviously, .
be a solution of the following Dirichlet problem with sufficiently smooth boundary functions
Theorem 2 Let and
, be sufficiently smooth functions
. Then the necessary and sufficient solvability condition for Neumann problem
(1), (2) has the form
If a solution exists
, then it is unique up to a constant and can be represented as
where is the solution of Dirichlet problem (15) with boundary functions , , , which satisfies the additional condition .
Proof Let a solution of problem (1), (2) exist and let be this solution. We apply an operator to a function and denote . Now we prove that the function is the solution of Dirichlet problem (15) with the additional condition .
From the properties of the operator
. By virtue of the following formula [9
one has for
We rewrite these conditions in a more convenient form. To do this we first consider the last two of them:
We multiply expression (19) by
and sum to (18). Then, making use of (17), we obtain
Further, by repeating this argument for all
, we get
Thus, if is the solution of Neumann problem (1), (2), then the function will be the solution of Dirichlet problem (15) with the additional condition . Note that, under the conditions of Theorem 2, the solution of problem (15) exists and it is unique (see, for example, ).
Next we find the conditions to the boundary functions
, which guarantee the equality
. Making use of the Almansi formula (see, for example, [11
], p.188) we write the solution of problem (15) as
where are harmonic functions in the ball Ω. Obviously, if and only if .
Substituting function (20) into the boundary condition of (15) and integrating over the sphere, taking into account the equalities
we get the system of equations
and . The matrix of this system is matrix A, defined by (14). As we noted above, . By reducing to the Vandermonde determinant, it is not hard to find the value of this determinant; one has .
Making use of Cramer’s rule, we find
from the above system of equations:
is the determinant of the following matrix
where determinants , , are defined above. Therefore, the equality holds if and only if . But by the definition of , this condition is equivalent to (16).
Thus, if the solution of the considered Neumann problem exists, then necessarily condition (16) holds.
We now prove the converse, i.e., if (16) holds, then the solution of the Neumann problem exists.
be the solution of Dirichlet problem (15). If condition (16) holds, then
and we may consider the function
and prove that this function is in fact the solution of the Neumann problem.
Indeed, after changing of variable
, the last integral can be written as
In the subsequent discussions, we use formulas (17) and (21) and assume that
. So, we have
Further, for the second derivative one has
Using the same argument, we have for any
Therefore, finally we have by induction