Bounds for the Derivatives of the Solution of the Neumann Problem

The method of the present paper was first developed in connection with the Dirichlet problem,1 which is based on the idea of the hypercircle in function space introduced by W. Prager and J. L. Synge.2 The present work differs from that of Prager and Synge in that they were interested in obtaining bounds in the mean square sense for elastostatic boundary value problems, whereas we have as our present objective the determination of bounds at a point for the derivatives of the solution of the Neumann problem. In the present paper, it is assumed that the hypercircle has already been found. This in itself is a rather difficult task but has been carried out for the Neumann problem by Synge in a previous paper.3 However, once the solution has been located on a hypercircle the remainder of the work is relatively simple. There are certain weaknesses in the method, namely, that the method does not apply to a point on the boundary except in very special cases,4 and that as the point at which bounds are being sought approaches the boundary the bounds become progressively weaker. Hence, we shall restrict the present work to apply only to point’s interior to the domain of definition of the problem