In this paper we are concerned with a distributed optimal control problem governed by an
elliptic partial differential equation. State constraints of box type are considered. We
show that the Lagrange multiplier associated with the state constraints, which is known to
be a measure, is indeed more regular under quite general assumptions. We discretize the
problem by continuous piecewise linear finite elements and we are able to prove that, for
the case of a linear equation, the order of convergence for the error in L2(Ω) of the control
variable is h |
log h | in dimensions 2 and 3.