  The ordinary Poisson brackets in field theory do not fulfil the Jacobi
identity if boundary values are not reasonably fixed by special boundary
conditions. We show that these brackets can be modified by adding some surface
terms to lift this restriction. The new brackets generalize a canonical bracket
considered by Lewis, Marsden, Montgomery and Ratiu for the free boundary
problem in hydrodynamics. Our definition of Poisson brackets permits to treat
boundary values of a field on equal footing with its internal values and
directly estimate the brackets between both surface and volume integrals. This
construction is applied to any local form of Poisson brackets. A prescription
for delta-function on closed domains and a definition of the {\it full}
variational derivative are proposed.
