<DOC>
<DOCNO>
731
</DOCNO>
<TITLE>
upper and lower bounds for the solution of the first
biharmonic boundary value problem .
</TITLE>
<AUTHOR>
diaz,g.b., and greenberg,h.g.
</AUTHOR>
<BIBLIO>
j. math. phys. 27, 1948, 193.
</BIBLIO>
<TEXT>
  let w(x,y) be a solution of the boundary value problem
where r is a plane
domain with the boundary c .  the authors obtain upper and
lower bounds for, the value of w at a point in r,
by a method which is applicable to many other problems .
  if u is a function satisfying the boundary conditions
and v is a function satisfying the partial differential equation,
then the authors obtain by applying green's classical
identity and schwarz's inequality a pair of inequalities
of the form where .
  together with the function w the authors consider a
function the solution of the boundary value problem
on c, and in
analogy with the functions u and v associated with the function
w a pair of functions and associated with the function
.  in the expression for derived from green's
classical identity appears an unknown line integral containing
the values of w and on c .  but the same line
integral appears also in the expressions for
to which the above inequalities are
applicable .
  in this way the authors obtain two inequalities of the form
where b and b', respectively, are approximate
values of .  in order to improve these bounds
one may add to u a linear set of functions and to v a
linear set of functions and then minimize h(u-v) in order
to determine the coefficients of the best linear combinations .
if the sequences and are complete in a certain sense
defined by the authors the approximations will converge to
the value .
</TEXT>
</DOC>
