  We show a weighted version of Korn inequality on bounded euclidean John
domains, where the weights are nonnegative powers of the distance to the
boundary. In this theorem, we also provide an estimate of the constant involved
in the inequality which depends on the power that appears in the weight and a
geometric condition that characterizes John domains. The proof uses a
local-to-global argument based on a certain decomposition of functions.
  In addition, we prove the solvability in weighted Sobolev spaces of div(u)=f
on the same class of domains. In this case, the weights are nonpositive powers
of the distance to the boundary. The constant appearing in this problem is also
estimated.
